In a photographer’s studio in north London, sitting at a wooden table with mugs of tea, two friends are having a chat. They’ve discussed food and clothes, but now they’re on to their actual friendship: the reason why we’re here.

“I think before this podcast,” says Miquita Oliver, “a lot of people who were aware that Lily and me have a friendship would be basing it on pictures of us leaving the Groucho pissed when we were 23…” She looks at Lily Allen, who is laughing quite hard.

“When we used to leave the Groucho,” says Allen, “there would be these paparazzi outside… And Miquita would always grab a copy of the Evening Standard, to try to make herself look more studious!”

Oliver (also laughing): “Just so you know, I’ve been reading the paper, while they all got fucked up!”

Allen: “I’m going to get a picture of you doing that and frame it.”

“Never,” says Oliver, “get any framed photo of us leaving the Groucho. Ever.”

Lily Allen, pop star and actor, and Miquita Oliver, TV presenter, are not 23 any more. They’re 39 and 40, respectively, with much calmer night lives than their Groucho years, though their mischief-making and mickey-taking attitudes remain. For evidence, we have their new twice-weekly podcast, Miss Me?. On paper, it’s not so different to other chum-pods out there, meaning it’s two famous people, having a nice time, in front of mics. Initially, Miss Me?’s USP appeared to be that they live in different places, and their lives are at different stages – Allen lives in Brooklyn, New York, with her husband, actor David Harbour, and her two daughters, Ethel and Marnie; Oliver lives alone in east London – but other than that, there wasn’t much to distinguish it.

But Miss Me? is different, and the difference is their friendship. Lily and Miquita have known each other all their lives. “We’re not just Made in Chelsea mates,” says Oliver. “We’re more like sisters,” says Allen.

The two women grew up together in west London; their mums are really close; their families are intertwined. And it’s this relationship that lifts Miss Me? above all those other pods. It arrived ready, confident, fully formed, and it’s improved from there; as the Observer’s audio reviewer, I can say from experience that this is rare. You could tell from the first listen that it would be a hit, and it is (it’s regularly in the Top 10 of the podcast charts).

The way that Oliver and Allen talk and spark off each other is not only hilarious – they’re both naturally very funny – but also intimate. Over the 13 weeks they’ve been making the show, they’ve told hair-raising anecdotes and revealed emotional truths, simply to make each other laugh. They have wide-ranging interests: there’s a lot about early 2000s style and silly romcoms, but Oliver also recently waxed lyrical about John Betjeman’s classic suburban TV documentary Metro-land and Allen, having read Jonathan Haidt’s The Anxious Generation, is trying to stop her children’s mobile phone use. Still, as you might expect, the stories that get the most attention are those when they’ve stepped on each other’s toes romantically: Oliver said she’d snogged one of Allen’s boyfriends while Allen was in another room; Allen admitted to sleeping with a pop star that Oliver had “claimed” as hers.

Sometimes the stories are more serious, such as when Allen discussed the 2017 Finsbury Park mosque attack. When the perpetrator was caught by police, he had a list on him of names that included Jeremy Corbyn, Sadiq Khan and her: “It’s one of the reasons I left this country and moved to New York,” she says.

In one of the two weekly shows, Listen Bitch!, listeners ask questions on a theme, and these have proved revelatory. Not only are the questions excellent – a question on the show with a “sabotage” theme resulted in Oliver confronting Allen about sleeping with “her” pop star – but they come from all over the world (Spain, Argentina), as well as the UK.

Oliver and Allen also get political, although, as the podcast is put out by BBC Sounds, this is carefully edited: a recent episode had them dancing around their opinions of the candidates for the UK’s general election, and, in an early show, Allen’s comments about the Met Gala were edited. “I was talking about reproductive rights, and Roe v Wade,” she tells me, “and I said that, at the Met Gala, there were enough people in that room to have really moved the needle about that, and the legal department was like, ‘That sounds like a call to action’, and it had to be edited out.”

She thinks. “I still don’t understand the guidelines.”

“No, but I don’t think it’s our job to dissect them,” says Oliver. “It would be ridiculous for us not to talk about important shit that’s going on that we care about.”

I say I’m surprised their podcast is on the BBC, because of their outspokenness, and they say they shopped it about, and the BBC was the only platform that let them do what they wanted. “They’re really supportive,” says Oliver.

Anyhow, their chats aren’t exclusive or excluding, which is amazing because they’ve led privileged lives. Not in terms of money – their families were often skint – but because they’ve grown up around creative people and had access to artistic adventures that most of us could only dream of. Allen’s mum is the film producer Alison Owen, and her dad the actor Keith Allen (they split up when Lily was four). Oliver’s mum is singer turned TV chef and presenter Andi Oliver, whose best friend is Neneh Cherry. Such connections mean that, on the podcast, Oliver will casually mention that Lily Savage (Paul O’Grady’s drag persona) was the first person to ever do her makeup, or Allen will chat about being friends with Elton John, and then feeling resentful when he stopped getting in touch. Somehow, because their tales are honest, with themselves as the punchline, they make everything relatable.

They’ve been told, a lot, that their show is nostalgic, which I don’t quite understand. “I don’t think it’s necessarily nostalgic of a certain time,” says Allen. “The language and the dialogue that we have with each other reminds people of their relationships with their best friends, maybe when they were young. It’s nostalgic of that.”

“It’s also nostalgic because people used to see me on TV when they were younger,” says Oliver, “in a formative time in their lives. Usually hungover.”

She’s referring to the years she presented Saturday morning TV on Channel 4. Oliver was only 16 when she, alongside Simon Amstell, started interviewing pop stars on the brilliantly sarky yet celebratory music show Popworld. She retains a more journalistic edge than Allen and regards broadcasting as a service, especially weekend TV: “When I do Sunday Brunch, I’m like, I’m waking up the country!” Her production instincts are strong: “Miss Me? has got to be the truth. Otherwise, what’s the fucking point? That’s why people like it, because it’s just the truth. And the truth is messy.”

Broadcasting doesn’t come quite so easily to Allen, who can be wary and self-critical (“I thought about Miss Me? the same thing that I always think, which is people are going to hate it”). In the past, she’s been burnt by the press. From the moment she broke through as a pop star, aged 21, with Smile getting to No 1, she’s been a relentless focus for the tabloids, with paparazzi following her every move in her 20s. I idly mention that, at a previous interview, she had two phones, and she says that was because of hacking. For several years, she’d get what she assumed were prank calls from kids, but realised, after a while, that they were from journalists. They would call repeatedly until she dropped the call, because then her phone would go to voicemail and they could get into her messages. (Oliver: “Oh my God, I didn’t know that!”)

Even now, Allen gets a lot of unwanted press attention: stories from the podcast go straight on to the Daily Mail online. Just before the show launched, she made a wry comment in a Radio Times interview about her kids wrecking her pop career that was splashed all over the papers, and it freaked her out. Not only because she felt exposed, but because the quote, and therefore her vulnerability, had come about because of Miss Me?. “If I’m suddenly Daily Mail fodder again, I don’t want to feel like you’re a part of that,” she says to Oliver.

But, actually, since the show started, she’s become more relaxed (you can hear this as the podcast has gone on), because Miss Me? allows her to be herself, without journalistic mediation: “With this show, people can be like, ‘Oh, hang on, she wasn’t attention-seeking. This is actually how she is as a human being. I can see it and hear it in the show.’”

They’re both protective of Miss Me?’s format, which is: they have to be in different places and see each other on FaceTime, so it’s really like they’re catching up; they decide on the topics they’ll talk about; and no guests. Allen: “I don’t want to have to spend my life, when I’m socialising, thinking, ‘Would this person be good on the podcast?’” Though just the other day they broke their rule and did a special interview episode with Billie Eilish. Eilish, 22, has an excellent new album, Hit Me Hard and Soft, and her people got in touch to see if she could be a guest. Initially reluctant – “that’s not what we’re about!” – they thought about it and said yes (obviously). They both loved Eilish, who was as honest as they are, speaking on the show about how hard it has been for her, as a young famous person, to find and hold on to real friends.

“I felt really protective about her, it was so weird,” says Allen. “Because Miquita’s background is probing pop stars. And my job is being the pop star, and I’m also a mother. Billie is closer to my kids’ age than she is to mine. So Miquita would say, ‘What do you mean by that?’ to Billie, and I was like, ‘She doesn’t mean anything. She does not want to answer that question.’”

Though both Oliver and Allen are clearly enjoying Miss Me?, even five years ago it would have been impossible for them to make. Partly, it’s because Allen has moved to the US that it works (“I wouldn’t have done it if I was in the UK,” she says), but also, they’re both currently in a good place, which is vital. Separately, they’ve worked hard on their own lives, and because of this their friendship has blossomed into something resilient and collaborative enough for a podcast.

Over the years, they’ve had some tough times. Oliver’s late 20s, after she quit T4, were difficult. She’d joined the show – a comedy/celebrity youth programme – straight after Popworld, and loved hosting with other talented presenters, including Alexa Chung, Nick Grimshaw, Steve Jones. But Jones left in 2010, and she felt the show was changing in a way that she didn’t like – “Before, even when we were dressed up as crabs, there was an honesty about it, we were honest crabs” – so she handed in her notice. It was her decision, but she still felt fired, “or maybe discarded”; she’d been a regular Saturday TV presenter for 10 years. Without plans in place, she coasted, career-wise. And then, in 2012, she had to declare bankruptcy.

“It was months of letters that I was hiding, pretending they didn’t exist,” she remembers. “There was this bureau in my flat full of letters saying: ‘You need to pay your tax.’ And then the letters started saying ‘bankruptcy’, and still I kept ignoring them.” She didn’t discuss the situation with anyone, because she was so scared and ashamed. Eventually, the bill got to £170,000. To sort it out, she went to a big glass office where she was interviewed by HMRC officials – “they tear you apart” – and was told that she had to give up her two-bedroom flat and go and live with her mum. “I had, like, a fiver, but I thought, OK, at least I’m safe. It was interesting to realise that I didn’t need that much.”

She pauses. “I usually work quite well after disaster. That’s usually my MO.”

Her latest comeback started around five years ago, when she felt she was drinking too much and quit for six months (she drinks a little bit now). She started making some programmes with her immensely charismatic and talented mum, such as BBC Two’s The Caribbean and Stirring It Up, a regular friends-round-for-a-meal podcast. She also set up a skipping school, Ropes, and has many other projects in the pipeline. “I’m planned and strategic with work stuff, and Lily’s not really,” she says. “This summer, I’m taking a leaf out of her book and just going with the flow… But there are shows I want to produce, and I want to write stuff, and make TV shows that I want to present.” She’s also trying to manifest a suitable boyfriend. She wants kids (she’s gathering positive stories about older mothers) and is imagining “an artist in south London” for herself. “Someone who doesn’t like Instagram and wears good tracksuit bottoms.”

Allen’s difficult times have been well documented: they include the stillbirth of her son, George, in 2010, a divorce from Sam Cooper, the father of her two girls (confirmed in 2018), and a terrifying encounter in 2015 with a long-term stalker who got into her bedroom and was only scared away because her then-boyfriend was there. She was accused of racism in 2013 because her Hard Out Here video featured black dancers in fewer clothes than her, mocked for passing out drunk at Notting Hill carnival, and slagged off for raising issues around refugees by going to a migrant camp in Calais. “I think that voicing my opinions and views as a woman, it made me a lightning rod,” she says. “When women demand to be heard – Meghan Markle is the same – you become like a cartoon. And… I don’t know. Maybe I’ve got a weird little punchable face.”

She moved to the US four years ago, and married David Harbour in Las Vegas in September 2020. They live with her daughters in a beautiful house in Brooklyn. Her life in the US is calmer than her life was here – partly because she isn’t recognised, but also because she’s been sober for five years.

“I’m a much more connected and present parent, which is amazing,” she says. “I got a text from Ethel yesterday and she was very clear: these are the things I’m feeling, I’m sad, I’m upset because of this. And I was, Oh my God, I’m winning at life! She’s naming her emotions and she’s asking me for help. My relationship with my kids specifically is much better since I got sober.”

The part of sobriety she finds hard is around creativity. “Drugs and alcohol were very good at drowning out my inner critic,” she says. “When you take cocaine you think all of your ideas are brilliant and need to be shared.” She’s been working on songs – “I have about 100 of them!” – but doesn’t feel like they’re any good, so she’s called time on that side of her career for a while. Next summer she is due to star in Hedda Gabler, in Bath, so she wants to enjoy this summer with her daughters.

“I can really miss those extreme highs,” she says, “like I miss being on the Pyramid stage at Glastonbury and rushing my tits off. I know that I will never experience that again, and there’s a sadness that comes with that. But I am level. I’m no longer having really terrible, long periods of time of absolute despair.”

So they’re both in a good place, and their relationship with each other is, too. This was not always the case. “Not at all!” laughs Oliver. Allen: “The way that we behaved with each other in our late teens and early 20s, this generation would be like: ‘Oh my God, that’s so toxic!’” They’d be jealous of each other, they’d compete with each other, they’d argue and fall out for several months.

Now, they text each other and say sorry, or talk it out in the moment and try to understand. “The show’s been good for us,” says Oliver. “Because we have to have pure communication, which is great for any relationship.”

I wonder if that communication is extended to their mums, both fabulous women who can, we have learned from the podcast, get on their daughters’ nerves.

“Lil and my mum can go Tupperware shopping on holiday – which they did – for two hours,” says Oliver, “and come back and they’re in love with each other. That would drive me nuts. But me and Auntie Alison can talk about shit for hours… We’re both very aware of how brilliant our mothers are. I love that my mum never stops. She never stops. We have big plans, and she makes me feel like I can do anything. I love having a mother like that.”

Allen’s mum is currently in Brooklyn looking after Ethel and Marnie, because Allen is in the UK and Harbour is away working. “Our family really wouldn’t work without her stepping in like that,” says Allen. “She’s amazing with the kids.” She says she found it difficult when she was younger that her mum was less present because she worked so hard, but that “she taught me that being passionate about what you do is a gift, but it takes hard work and sacrifices have to be made. And she taught me how to make an impeccable roast dinner and spaghetti bolognese and trifle.”

Oliver is laughing again. “You know that our mums want to do a podcast now?” she says.

Allen: “They think they can do their own…”

Oliver: “They want to call it The Corrections…”

Allen: “That’s what we mean. We can never get away…”

• Miss Me? with Lily Allen and Miquita Oliver is a Persephonica production for BBC Sounds. New episodes launch Mondays and Thursdays