No shoes. Or none worth mentioning. What does Natalie Mendoza, a soaring Imelda Marcos, wear with the best stage costumes for years? With the kingfisher silk dress which has shoulders like shark fins, the cinched-in New Look frock covered in fat flowers, the gold dress that seems to be made out of gold coins? Terrible clumpers. In fawn.

That is the only real disappointment in Here Lies Love, David Byrne and Fatboy Slim’s speedy, rowdy show about the Philippine dictator’s wife. Here Lies Love does not wag its finger. It starts with celebration and cleverly turns it into condemnation. Alex Timbers’s heady production cajoles spectators into taking part in life as run by the Marcoses. Empty, showy, all glitter and no guts. Manipulative, crooning its spectators into dazzled obedience.

Half the audience looks down on the stage from galleries. Others are on a dance floor, through which some of the action moves on raised platforms, pushed by functionaries. Down there I was dazed by the glitter ball, the strobes and the massive sound. Then humiliated by being made to jump at the decree of those on the platform. Just as I was beginning to revolt at my acquiescence to these flashy dictators, the show began to change. Colour started to drain away with the onset of news of Marcos thuggery and debauchery. Amplified noise died down as the peaceful Marcos regime was peacefully overturned. And blare gave way to drums and a soft guitar.

This musical innovation is enhanced by the reshaping of the National Theatre taking place around it. I hate renamings. What has poor old Cottesloe done to have his name banished from the smallest auditorium, now called the Dorfman after Travelex founder Lloyd Dorfman, who gave the theatre £10m? Yet I love the effect. You can tell this is a building by Steve Tompkins of Haworth Tompkins, just announced as the winner of the Stirling prize for his re-creation of Liverpool’s Everyman theatre. There is his trademark red, a characteristic bare patch of brick, and the smell of fresh timber. Above all, this is the opening out of a boxed-in space, now more welcoming both to the river and the area at the back of the National. And to spectators. Walkways with glass panels will allow people to stroll above working areas, looking down on paint rooms and carpentry spaces, on the ropes from Treasure Island and a recumbent statue of Jesus Christ.

Stratford East has always maintained its gorgeous traditional scarlet and gilt, along with its onstage iconoclasm. The Infidel – the Musical is its best tribute for ages to Joan Littlewood’s theatre. It’s bawdy. The risks it takes are real. It has, without being po-faced, a healing purpose. Actually, there has not been anything since Jerry Springer – the Opera that targeted religious zealots so exuberantly. Unless you count The Book of Mormon, which I don’t, thinking it plenty rude and finely drilled but toothless.

David Baddiel’s tale of the Muslim bloke who discovers in middle age that he was adopted and is in fact Jewish is new to the stage but not entirely new, being based on the 2010 movie directed by Josh Appignanesi. Baddiel’s punchy lyrics lay acerbically and joyfully into Jewish and Muslim fundamentalism.

To be the hit it deserves to be, it needs a tummy tuck of about half an hour in the second half. It could also do with a few more memorable tunes from Erran Baron Cohen (brother of Sacha), whose perky pop score mixes rap and gospel and ballad. Still, this shiksa went off through the gleaming wastes of Westfield humming “I’m a Jew, I’m a Jew”, with at least two more numbers trapped in her head. A song set to the tune of Camptown Races has as its chorus “Judah! Judah!” A duet about circumcision, Less Is More, is sung by Muslim Mahmoud and Jewish Lenny, accompanied by slides of domes and towers. Melanie Marshall uses the lovely blast of her voice to hymn the sexiness of the burqa. Kev Orkian, who with mischievous cleverness bamboozled Britain’s Got Talent audiences about who he was, is spot-on as the hero with the identity crisis. Light and wry, his comic gift is as strong as his considerable voice.

Is musical theatre heading towards more exciting times? These shows – and the forthcoming Made in Dagenham – may be signs of that. Yet they should not diminish the sucking, squelchy, brassy, defiant, melting, zinging power of a big Broadway musical. Of which Gypsy (music by Jule Styne, book by Arthur Laurents, lyrics by Stephen Sondheim) is one of the very best.

It is in most assured hands with director Jonathan Kent, master of fleetness and the big effect. Apart from a pale performance from Kevin Whately in the admittedly hangdog part of rejected suitor, everything is in place. The ailing Vaudeville acts (with straw bonnets and panto cow) and bouncing burlesque grotesques. Dancing from Dan Burton that makes weight look like an alien concept. Lara Pulver subtly turning from plain to slinky. And Georgia Pemberton putting in a terrifying turn as the tot performer, her rictus smile as wide as her bloomer-clad legs.

As Momma Rose Imelda Staunton enters like a female Napoleon. Her walk is beetling; she hits the words of her songs like a terrier shaking a rat. She is the hellish quintessence of stage mother, a magnified version of any woman who has sapped her child’s energy by her own superior vitality and the force of her own need. She is also every aspirant American, every aspirant anyone, pushing herself out of a pokey place and her children into the spotlight.

Staunton gets the gusto, the ruthlessness and the pathos. As she sings with her future beau, the voice softens, and you see what she might have been if she hadn’t always been fighting. As she is left by the daughter she wanted to make a star, she shows not grief but defiance. Her Everything’s Coming Up Roses is an act of will, as she bites back anguish and a panic attack. A tiny figure on a huge stage. A big voice, and behind it gigantic shadows. One of the greatest of musical monsters.

• Here Lies Love is at the Dorfman (formerly Cottestloe), London SE1 until 8 January; The Infidel – the Musical is at the Theatre Royal Stratford East, London E15 until 2 November; Gypsy is at Chichester Festival Theatre until 8 November