"So, they cleaned up the neighbourhood..." Cy Coleman and Ira Gasman's cult musical turns the clock back on Times Square. Before Broadway was all Disney musicals and Lin-Manuel Miranda, before the Elmos and Elsas moved in for good, the stretch from 42nd to 47th was the seedy underside of New York. Pushers pushed. Hustlers hustled. Prostitutes stalked every street corner.

Inpsired by an arrest Gasman witnessed outside a theatre, The Life pulls together a string of street tales, all shot through with the spirit of Damon Runyon, whose short stories inspired Guys and Dolls. The Life's like a return visit 50 years later, and the Broadway it uncovers has a far darker heart.

It's a world that preys on the vulnerable, and pulls the ladder up on those that fall into its trap. Queen (T'Shan Williams) dreams of a better life with her boyfriend Fleetwood (David Albury), a Vietnam vet battling demons by night, but for now, she's stuck on the sidewalk. He turns his hand to pimping, less to save for their future than to satisfy his coke habit, scooping up New York newbies fresh off the bus. Mary (Joanna Woodward), just in from Minnesota, with hopes as high as the Empire State Building, is his first recruit.

David Newman's book is neatly structured, with two plots - one rise, one fall - running in tandem, but it's made by its characters and by Coleman's crack score. Really, it's a musical postcard, the sense of a place through song, and by the time the plot kicks in at the end, it has a lot to do in a short space of time.

No bother, Coleman and Gasman's numbers could be cabaret standards, their self-contained stories told with gut feeling and wit. "My Body" slyly asserts a woman's right to mind her own business, while "Easy Money" trills with the exhilaration of earning good cash for the first time. There's sass in this score, but underneath it, in its bass notes and darker numbers, something anxious and menacing. "Don't Take Much" presses home the way men call the shots; its low-key timbre sliding into a sour, chauvinistic sense of humour.

Veteran director Michael Blakemore, who launched the Broadway premiere 20 years ago, puts on a good show, but it is just that: a show. There's no sense of Times Square's reality on Southwark Playhouse's small stage. The danger of those backstreets, the dinge of those bars, none of that ever entirely comes through. It's all a bit, dare I say it, musical theatre: wise-cracking, high-kicking hookers and twitchy hoodlums chewing on toothpicks. Dress that up in the leathers and velours the late '70s left behind and, without every effort to ground it in reality, the whole thing trips into fancy dress camp. Beneath Nina Dunn's glossy cityscape projections, it all gets a bit I HEART NY.

Only Sharon D Clarke really shrugs that off as the hardened old-timer Sonja. She's tough as old boots, but soft as a shagpile – the patron saint of streetwalkers and, somehow, the spirit of dirty, old New York. She alone wears the weight of a life spent on the sidewalk – someone who's seen it all, done it all and can still summon up a smile. When she sings "The Oldest Profession" ("I'm getting to old for… etc etc"), she gives it all the exasperation, resignation and humour you'd expect of a woman who needs a calculator to tot up her tally. Clarke, astonishing, turns the final figure from a shrug to a snarl.

Cornell S John instils a genuine threat as the ever-present pimp Memphis, still as a snake charmer and proof that anyone with real power rarely needs show it, but David Albury's cokehead is too fresh-faced by half and, for all her country looks, Joanna Woodward never convinces us that Mary would do anything - yep, anything - to get ahead. There's always a hint of Home Counties about her.

That's true of the whole show, really; a great, unsung musical that, if it's to really nail its revival, needs a sharper sensibility than this - a staging that rather fluffs its feminist credentials with a quick flash of its leading ladies' tits. That's a mark of Carry On cheek, not Times Square's squalor and this neighbourhood never warrants a deep clean.

Click here to watch our exclusive video with the cast of The Life

The Life runs at the Southwark Playhouse until 29 April.