THEY’VE ditched plenty that made the musical of E. L. Doctorow’s Ragtime such a lavish success on Broadway: the stadium in which tiers of singing, spitting fans appeared at the mere mention of baseball, the vast grilles that kept immigrants trapped on Ellis Island, J. P. Morgan’s luxuriant library, a real car for Coalhouse Walker, a real engine for his fire-service foes, much more.

Never mind. Terrence McNally and Stephen Flaherty’s show exercises more than sufficient grip on a stage with no more decor than grubby mirrors at the back, a steel bridge above, and plain chairs doubling as cars and engines. Indeed, last night the piece seemed timely as well as arresting. As I drove home, hearing on the radio that I should stockpile food and water, one of its central topics didn’t feel fictional at all.

That’s urban terrorism. The time is the early 1900s. Of the three classes in the show, the WASPs are doing fine but the immigrants are battling with poverty and the blacks with racism. And Kevyn Morrow’s charismatic Coalhouse transmutes from a genial, genteel Harlem pianist with a taste for rag into a man who, after bigotry has destroyed first his car and then the mother of his child, loses another kind of rag on a scale all too familiar to us today. Think denial of justice, think frustration and obsession, think vengeance by suicide bombing.

Coalhouse’s tale criss-crosses that of sometimes generous, sometimes prissy upper-crust whites baldly called Mother, Father, Younger Brother and Boy. So does the other main story, that of the Jewish immigrant, Graham Bickley’s Tateh, who makes his fortune as a film pioneer after half-starving in a New York so Stygian he’s forced to keep his daughter on a rope, lest she’s stolen. There are terrible downs but also spectacular ups. That’s America circa 1905 and maybe even 2003.

Not for nothing does Houdini join the black leader Booker T. Washington and the anarchist Emma Goldman as one of several real-life characters. The capsule theme is the desperate need and fearful difficulty of breaking out of social straitjackets. And though the stage sometimes seems as crowded as a rush-hour tube, Stafford Arima’s production catches the restlessness, the turbulence, the dangerous excitement of an era when everything from transport to morality was in flux.

All right, one side of me sometimes found the rehearsal-room staging a bit stark. Another wished that Houdini was more active and less metaphoric, as I recall him being on Broadway. Again, the lyrics become sententious, especially when they are earnestly debating the pains of progress. But Maria Friedman’s Mother, the voice of white decency and all-American hope, has enough simple passion and sweetness to cope with Flaherty’s more soaring odes — and Flaherty has the skill to obey the show’s title and combine syncopation with melody.

Is the musical too American for London? That’s what I had feared. Last night a brisk, pacey, tuneful and touching premiere provided a clear answer: no.

Box office: 020-7369 1734