What's the Welsh for Les Mis? It may well be Tiger Bay. Daf James and Michael Williams' massive new musical, which charts the social tensions and economic pressures around Cardiff's docks at the turn of the century, is straight out of the Boublil and Schönberg playbook. It pulls together disparate, rivalrous characters from all walks of life into a rousing show of social solidarity.

At the height of the industrial age, Welsh coal made Cardiff's docks a destination, drawing money and people from all over the world. It led its owner, the Marquess of Bute, to the top of the global rich list, while the suburb named in his honour, Butetown, housed a heaving and diverse immigrant population. That history is still a source of local pride – Tiger Bay testifies to a long history of multiculturalism in Britain – even if the area has long since changed beyond recognition. Coal exports seized up, depression set in. It's mostly chain hotels and shopping malls today.

With riches comes change, and with change, disruption. As coal-dusted ‘donkeymen' shunt cartloads through the docks, aided by urchin'd 'waterboys' greasing the tracks, they're pushed in turn by Noel Sullivan's sly, hard-nosed harbourmaster, Sean O'Rourke. Any threat of a strike is kept at bay by that of unemployed immigrants ready to step in, and racial violence routinely spills into the streets. James' score heaves and hos with the rhythm of work, but splits into cacophonies whenever chaos erupts.

It this midst, an unlikely partnership forms between Themba (Dom Hartley-Harris), a refugee widowed by the Boer War, and his scampish waterboy Ianto (Ruby Llewelyn). Sullivan conveys O'Rourke's central contradiction: his economics press people down but they lift the place up. Romantically, he's caught between his spirited shop girl fiancé Rowena (Vikki Bebb) and the dock prostitute Klondike (Busisiwe Ngejane), while his managerial callousness evaporates in his own boss's presence. As the Marquess, John Owen-Jones, voice pure as spring, conveys the idealism of a man in an ivory tower, mourning his late wife and pining for a lost son.

You can likely see what all that tees up, and for all the skill with which Williams weaves narratives together, Tiger Bay can feel like it's running on rail tracks. It's all-encompassing and industrial, too vast to move perhaps, but always engaging over three hours, even when its second act struggles for propulsion under narrative strain.

Every new musical faces the hurdle of novelty. The best scores have songs you feel you've sung forever. If that's true of Tiger Bay, it's because you have done. Local boy composer James ships classic tunes in wholesale. "Down to the Docks" is trilled to "Into the Woods," while the enticing melody of "Joanna" seeps through the whole score. There's so much Sondheim, he's surely due royalties, but echoes of Oliver and Les Mis, maybe more, abound.

If it can feel like a round of Name That Tune, there's justification. Repeated tunes suggest the past rolling around – and Tiger Bay's troubles look a lot like our own – but they also allow James to break through familiar tunes with bursts of elsewhere. Arabic prayer and African drums add new textures and riches, and Melly Still's staging brings cartloads of atmosphere to Anna Fleischle's monolithic grey set. Wales being Wales too, is in fine, fine voice – worth going just to hear the people sing.

Tiger Bay runs at the Wales Millennium Centre until 25 November.