According to the producer, Tanya Link, Miss Atomic Bomb is the result of a five-year development process. It sure feels like it. Everything about this keen-to-please ersatz American musical (by two British writers, Gabriel Vick and Alex Jackson-Long, and Adam Long of the Reduced Shakespeare Company), played out in the mushroom cloud of the nuclear tests of the 1950s, is strenuous, efficient, well drilled and designed – and curiously flat, unfunny and flair-free.

The songs have no zing or provenance and sound as though they've been made on a computer, or compiled on a "how to write a song" study course; there's not a single moment of lift-off, melodic surprise or harmonic twist from start to finish. Aha – maybe that's a result of the radiation emanating from the Nevada desert.

A soldier on the run, Joey (Dean John-Wilson), hides out on a sheep farm in Utah where Myrna (Catherine Tate), a twangy-voiced hillbilly with ideas of being a fashion designer, and her best friend, Candy (Florence Andrews), dream of California while under threat of having their trailer repossessed.

Meanwhile, down on the Las Vegas strip, there's a cabaret going on at the Golden Goose hotel, a cheap tourist trap that Lou Lubowitz (Simon Lipkin) is desperately trying to upgrade while hounded by Mafiosi and bugged by a shock-haired potty professor (Stephane Anelli) from the nuclear programme. As you'll guess from a close reading of the title, all parties converge on a Vegas beauty pageant – where there are three sad left-overs from Gypsy (a lesbian, a whore and a drag act) in search of "a real good time" – and I won't spoil your lack of interest by revealing who wins.

There's some decent writing around the characters of Lou and Myrna, and Lipkin and Tate make the most of it, which ain't all that much. He pulls a rabbi out of a hat (oy-vey) and gets shot in both feet, while she rabbits on in a spat and knocks up a little fur and diamante number for Candy, just dandy. Their futures conjoin.

Adam Long co-directs with choreographer Bill Deamer, whose witty dance routines leave no musical theatre cliché unexplored but whose spatial ingenuity does at least expose, possibly for the first time, the potential of this venue for middle-scale musicals. Andrew Lloyd Webber, sitting in the stalls on opening night, must have thought about how he could re-launch Aspects of Love here, say, or a smart revival of The Boy Friend.

As usual, everyone is over-miked, and the sound is pretty awful, though the upside of not being able to see Richard John's tight little band is the remarkable design by Ti Green, which has a rainbow-like curvilinear metal bridge spanning a receding perspective to the desert and sky; the show's top moment is an ensemble apprehensive chorale in protective eye shades considering the distant cloud and the end of civilisation. The music, predictably enough, doesn't rise to it.

Miss Atomic Bomb runs at the St James Theatre until 9 April.