Although Broadway has a theatre named after the reviewer Walter Kerr and the English critic Nicholas de Jongh once had a play put on, theatre and theatre criticism tend to be inimical disciplines. So one of the distinctions of Forbidden Broadway, which has been running, off-Broadway, on and off since 1982, is that it is a show that consists wholly of critiques of other shows.

And these are not the sort of pseudo-reviews – mild jokes about a performer's ego or CV – that are cracked at theatre awards ceremonies to prove what good sports the stars are. Putting satirical new lyrics to songs from living musicals, Forbidden Broadway is relentlessly, cleverly, breathtakingly funny, most resembling a theatrical equivalent of Spitting Image, with the grotesques created from acting rather than latex. Cameron Mackintosh morphs with the character of the Engineer from his production of Miss Saigon to become the Producer, a preening egomaniac who belts out a rewrite of the Engineer's big number, The American Dream, which claims to explain how Sir Cameron's allegedly slick but shallow theatrical tricks "make the Americans cream". And Sam Mendes, who directed Charlie and the Chocolate Factory, would also be advised to wear a disguise or a brave smile if he turns up at the Menier Chocolate Factory during the run. The second of London's two big-budget Roald Dahl musicals is found wanting, having "no inspiration/ this pale Matilda imitation".

But those numbers are gentle compared with the finale. There, in the lyric that has been least altered, "representatives" of various big producing corporations sing the Nazi anthem from Cabaret with the chorus changed to "Broadway belongs to me!"

An institution in New York theatre, the show has been available in the UK mainly as Broadway cast recordings, although David Babani, artistic director of the Menier Chocolate Factory, tried twice (in the West End and at the Menier) to make it work in translation.

The usual theory about the franchise's relative lack of success in London is that the comedy depends on recognition, with the punchlines requiring a dustbin full of ticket stubs, ideally – from Broadway rather than the West End. But the current London version (or "edition", as Forbidden Broadway prefers) is, under the supervision of creator-writer Gerard Alessandrini and regular director Phillip George, deliberately tweaked so it mostly features shows that originated or are running in this country.

Even so, some sketches in the London edition still seem aimed more at West 46th Street than Shaftesbury Avenue: a duet in which Chita Rivera and Rita Moreno, early stars of West Side Story, lament the public tendency to confuse them is typically witty ("My name is Chita and not Rita!" to the tune of West Side Story's America), but may struggle to find a sufficient number of Brits who recognise the subjects enough to mix them up. Another skit, which turns on the odd combination of earnestness and age-inappropriate eroticism in the cabaret act of Mandy Patinkin and Patti Lupone, also feels rather American-friendly and is not made much less Brit-hostile by introducing the male caricature as "Mandy Patinkin from Homeland."

But just as readers can get the jokes in a Craig Brown parody of Jeffrey Archer's prose without ever having read one of the ex-con's novels, so Forbidden Broadway can either console you for having sat through some of these musicals or provide delighted relief that you haven't. Despite never seeing the Irish musical Once, I loved Forbidden Broadway's vicious skit about it.

Alessandrini and George have a lethal eye and ear for the weaknesses of any popular piece. Most often, they are funny about the earnestness of big-budget musicals, ripping into the patronising politics of Miss Saigon and The Lion King. But in the case of a target that is itself genuinely humorous, such as The Book of Mormon, the writers reverse the blade and send up the alleged billionaire smugness of the show's creators, Matt Stone and Trey Parker.

Stone, Parker and their musical collaborator Robert Lopez are also accused of sloppiness in fitting words and music together. That's one of several scholarly gibes, such as that Bring Him Home from Les Misérables becomes Bring it Down, as a performer struggles with a song that it is in the wrong key for him. And just in case any audience members associate that joke with The Song That Goes Like This in Spamalot, along comes a song suggesting that Eric Idle got that idea from Forbidden Broadway.

Revue formats of this sort demand performers who are both impressive at impressions and able to switch at speed between costumes and styles. And the Menier production is lucky to have Ben Lewis, Damian Humbley, Anna-Jane Casey and Sophie-Louise Dann. Like the men, the name-hyphenated women (who may one day qualify for their own Alessandrini parody – "I'm Anna-Jane not Sophie-Louise") having performed in many of the projects being mocked. Humbley is a Stephen Sondheim specialist (including the Menier's Merrily We Roll Along), while Lewis is an Australian singing star, most familiar in the UK from the DVD of Andrew Lloyd Webber's Love Never Dies.

To my surprise, having attended with the fear that in-jokes are thin jokes if you haven't been in, this is the most consistently funny show I've seen since One Man, Two Guvnors and, although its audience appeal may inevitably be narrower, Forbidden Broadway deserves an extended presence at the scene of some of the theatrical crimes it so enjoyably satirises.

• Forbidden Broadway is at the Menier Chocolate Factory until 16 August.