Think of Mamma Mia, whose format Girl from the North Country shares, and then think again. Long and hard.

The musical at London’s Old Vic, which has attracted rave reviews, is ten times more interesting, a hundred times more sophisticated than that Greek-island romance. How could it be otherwise when the songs that pepper the story are all by Bob Dylan, whose worst ditty makes ABBA’s best seem terminally unimaginative and bland?

How otherwise when that story is told by Conor McPherson, the Irish dramatist who won an Olivier award for The Weir and Tony nominations for Shining City and The Seafarer?

Mark you, there’s something oddly disjunctive about a show that is in large part a tribute to Dylan’s edgy muse. Girl from the North Country is set in Duluth, Minnesota, yet its action occurs almost a decade before the composer was born there. His interpolated songs never take the story forward and don’t always follow logically on what the characters have just said.

But does that matter? No, because they catch the flavor of a play that occurs in 1934 and, often, the unspoken fears, longings, regrets, woes of men and women hit by the Great Depression.

The place is a guesthouse whose owner, Ciaran Hinds’s Nick Laine, is about to celebrate a Thanksgiving that is anything but. His wife, Shirley Henderson’s Elizabeth, has run half-mad, thanks largely to the death of a daughter she blames on him. His son is a would-be writer and a drunk who thinks that the all-too-rare jobs on offer are beneath him.

His adopted daughter Marianne, a black girl in a racist society, is pregnant, but not by the elderly but prosperous shoe salesman he wants her to marry. The bank is about to seize all his possessions, house obviously included, while his hopes of opening a hotel elsewhere with his lover have been dashed, since she too turns out to be penniless. And he has, he says, “no soul.”

But Nick is just one of the play’s many losers and loners. These include Arinze Kene as a black boxer, en route to Chicago in forlorn hopes of returning to the ring after three years in prison, and Stanley Townsend as a bankrupt businessman trailing an angry wife and a mentally retarded son.

They wrangle, they talk of destitute Americans living in tent cities, they mourn the rise in the suicide rate, they are always fending off despair and, touchingly, displaying what resilience they can. “What you wanna waste your life away here for?” asks Nick of the mistress whose long-expected inheritance has been gobbled up by lawyers. “I gotta waste it somewhere” comes the answer.

And they sing, sometimes directly at the microphone, sometimes backed by a chorus of cheaply dressed houseguests, always to the drums, cymbals and guitars which McPherson, who also directs, has brought onto a near-bare stage.

They do so with the full sanction of Dylan, whose enthusiasm for the production seems to have been markedly greater than his desire for a Nobel Prize. As his agent wrote to McPherson: “Bob loves this and you can use whatever you want, however you want. You can do whatever you like.”

Accordingly, the dramatist has used some 20 songs from all periods and artistic phases of Dylan’s lengthy career, but seldom the obvious or famous ones.

Yes, we hear “Like a Rolling Stone,” which is especially apt because the lyric includes “how does it feel to be on your own, with no direction home, like a complete unknown, like a rolling stone.” But there’s no “The Times, They Are A-Changin’,” which is equally apt, since these people feel trapped in a wretched present unlikely to be changed even by the “strong man,” the American Hitler, one worryingly seems to crave. Even Dylan aficionados may not be familiar with “Idiot Wind,” “Jokerman,” and “Señor (Tales of Yankee Power),” all of which are banged powerfully across the footlights.

Yet there’s reason for all of them, as there is for Dylan’s 1979 “Slow Train Comin’” which embodies the frustrations of the boxer and the fraudulent Bible salesman who sing it. Likewise for “I Want You,” which starts by invoking guilty undertakers and lonesome organ grinders, yet goes on to embody the passions that Sam Reid as Nick’s drunken son and Claudia Jolly as his departing girlfriend can’t otherwise fully express.

Again and again Dylan’s quirky poetry rises to the emotional occasion, reflecting inner and sometimes subconscious emotion or—as in the case of Señor (“Señor, señor / Can you tell me where we’re headin’? / Lincoln County Road or Armageddon?”)—asking questions about America that still seem pertinent.

It all adds up to a refreshingly, indeed astonishingly original occasion, put across by a cast which, though British, has felt its way into all-American history.

Just about everyone is worthy of mention: Sheila Atim as pregnant Marianne, Jim Norton as the shoe salesman haplessly trying to escape old age, Ron Cook as a morphine-addicted doctor who steps forward to become the show’s melancholy commentator.

But there are standout performances from Hinds, whose Nick dominates the action and radiates the gritty stoicism of a man said to be “a man tryin’ to run through a wall real hard”, and Henderson’s Elizabeth, whose squirming, writhing, slithering body and voice combine to express fury, contempt and an unsettling slyness.

Will that hermit, Bob Dylan, make the trip to the Old Vic he almost failed to make to Stockholm? He should do so, if only secretly. He’ll find his heart as well as his music transformed into triumphant theater.

Girl from the North Country is at the Old Vic, London, until Oct. 7. Details here.