Guys and Dolls has gone north – 90 blocks north to be precise. Michael Buffong's all-black production, the first in the UK, follows in a venerable tradition that dates back to Billy Wilson's legendary 1976 Broadway version and reportedly stems from Frank Loesser himself, but Talawa's is the first to relocate the iconic musical uptown – from Times Square to Harlem.

It makes sense. Damon Runyon's original stories may have revelled in the cultural melee of Midtown, where high-flying New Yorkers crossed paths with its low-lifes, but when it comes to celebrating the city's seedier side, Harlem was happening in the mid-1930s. As the Great Depression set in, unemployment there skyrocketed, and a thriving black market bubbled up into being. Ideal conditions for Nathan Detroit, Sky Masterson and co. to shoot their craps and while away their days wooing women.

The shift opens up musical possibilities too, and Simon Hale's orchestrations, following Wilson's approach, underpin Loesser's classics with jazz beats and inflections, while Kenrick Sandy (of hip-hop dance company Boy Blue) fuses traditional swing dance with traces of Harlem Jive. Even so, both seem a bit underpowered and Hale's re-arrangements only really kick in mid-way through the second half with Ashley Zhangazha's playful, counterpointed "Luck Be A Lady" and a gospel-infused "Sit Down You're Rocking the Boat".

But if you're going to shift Guys and Dolls to Harlem, you have to deliver Harlem. Buffong doesn't, at least not with any specificity. Soutra Gilmour gives us a lowly street corner, a long way from the bright lights and bustle of Times Square, but Buffong never taps into the neighbourhood's human dynamics. Where's the destitution of the Depression, which hit Harlem so hard? Where's the tension that would, come 1935, tip into a riot? Instead, we get the same collection of characterful sorts in colourful suits. It's all a bit – well, a bit 42nd Street.

That said, there's a grit to these gangsters you don't often see. Ray Fearon's numbskull of a Nathan, busting his maroon suit at the seams, could handle himself in a way most schmucks couldn't. He's no Nathan Lane, put it that way, and when Joe Speare's psychotic Big Jule flashes a gun to stack the odds in his favour, Fearon takes it in his stride with a shrug.

But a little really makes you crave more. These men are gamblers – compulsively so. They seek out any space, however dangerous or degrading, and bet on everything from dice to deserts. Gilmour floats the idea: her revolving stage looks a lot like a record, stuck in its groove as it goes round on repeat, and a neon sign overhead promises 'DRUGS' and 'PRESCRIPTIONS.' (The cure is always smaller than the allure.) Again, however, the action fails to follow through. Buffong gives us upright sharks, high-functioning even when they're rolling dice through the sewers, hostage to a 24-hour craps game. They keep their dignity intact throughout where, had they hit the bottom of the barrel, it would make a much sharper ending. That's Nicely Nicely's realisation, no? "Sit Down You're Rocking the Boat" is an awakening from the nightmare of addiction.

Indeed, addiction gets into every Guy and every Doll. Lucy Vandi's long-suffering Miss Adelaide can't stop herself spinning fictions about married life with Nathan and their, ahem, six kids, while Abiona Omonua's stoic Sarah Brown makes a habit of her soul-saving with the Salvation Army. It's all as unhealthy as everything else, and only Zhangazha's spick and sensible Sky Masterson – the shot in the show's arm – steers himself through life with some sort of control. He knows what he's doing and where he's going with each bet. If only the rest had such sense of self.

Guys and Dolls runs at the Royal Exchange until 27 January.