It was hailed in America as the “best musical of this century” and “the funniest musical of all time”. Now British audiences will get a chance to see what all the fuss is about as the multi-Tony-award winning The Book of Mormon finally arrives in the West End this month.

That's if you can get your hands on a ticket: the show, which tells the story of two young Mormon missionaries who are sent to Uganda to spread the word of Joseph Smith, has already sold out at London's Prince of Wales Theatre until June, despite its unknown cast and very American subject matter.

So just why is this brash, big-hearted tale of two clashing cultures the most anticipated arrival of 2013? The short answer is because it's written by Trey Parker and Matt Stone, the creators of long-running animated comedy South Park: “The power and popularity of South Park is massive in Britain,” says Sonia Friedman, whose company is producing the show in the West End. “That meant that there was a huge level of excitement and intrigue around the show before it even opened.”

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As you might expect, given Parker and Stone's involvement, The Book of Mormon is not your grandmother's musical. It can be almost breathtakingly crude, pushing the boundaries of taste with jokes about everything from female circumcision to child rape – the song “Hasa Diga Eebowai” features a litany of woes culminating in the rousing chorus “Fuck you God” – but it's also extremely funny (“Africa is nothing like The Lion King,” exclaims a bewildered, blood-splattered young missionary in one of thousands of jokes attacking poverty tourism).

It helps too that there is a kernel of truth at its core: when we first meet the fearsome Ugandan warlord, his name General Butt-Fucking Naked seems like little more than a rude joke: yet the character was based in part on Liberian warlord Joshua Milton Blahyi, who was better known as General Butt-Naked, and while his atrocities are played for laughs there is strain of real anger running throughout. Similarly, though the musical walks a fine line in its depiction of the apparent credulity of Ugandan villagers – seemingly swiftly won over by one of the young missionaries' cobbled-together stories of Star Wars and Lord of the Rings – it saves its most righteous disgust for the idea that two well-off white boys with Bibles can solve poverty with prayers.

Some of those jokes are wince-inducingly close to the bone as Parker, Stone and their composer Robert Lopez, creator of the much-praised puppet musical Avenue Q, make cracks about Aids and indeed about aid and take pot-shots at everything from The Lion King to Bono ('I am Africa! Just like Bono! I am Africa!').

Most of all there are jokes about Mormons. Jokes about their beliefs, their religion's origins, their founder, even their refusal to drink coffee.

“I believe that in 1978 God changed his mind about black people,” sings the devout Elder Price during “I Believe”, a show-stopping number outlining some of Mormonism's more outré tenets including the belief that God “lives on a planet called Kolob” and “the Garden of Eden was in Jackson County, Missouri”.

Yet underneath the mockery The Book of Mormon, like South Park, is a very moral show. Stone has described it as “an atheist's love letter to religion” and it has a surprisingly tender core. We might laugh at the devout Elder Price and his companion, the bumbling, love-struck Elder Cunningham, but we still hope that they and young Ugandan villager, Nabulungi, who dreams of discovering a utopia in “Sal Tlay Ka Siti”, find love and happiness in this strangest of situations.

“Its appeal lies in the fact it's both a spoof and utterly sincere,” says Ben Brantley, the chief theatre critic for The New York Times, and the man who called it the “best musical of this century”: “It's cool (as a satire) and totally square (as a hymn to the old-fashioned musical). It has it both ways with gleeful zest.”

As a formula it has paid dividends. The Book of Mormon won nine Tony awards including Best Musical in 2011 and continues to be one of the hottest tickets in New York long after the celebrities who crowded into the seats on opening week have gone. Indeed, so great is its popularity that the Broadway website now carries a warning against fake tickets, following a number of incidents where people were conned into buying seats for an already sold-out show.

Nor is its success limited to New York. It's currently packing theatres in Chicago as part of a widely praised national tour, and there are strong rumours of a movie to follow (asked who he would cast recently, Parker insisted he wanted pop prince Justin Bieber as Elder Price).

Even the Mormon Church has been, at least partially, won over: Mormons who saw the show on Broadway told USA Today it was “incredibly sweet” and praised Parker and Stone for “treating us with affection” and “doing their homework”. An official spokesman for the Church remarked simply: “The production may attempt to entertain audiences for an evening, but The Book of Mormon as a volume of scripture will change people's lives forever,” a statement Stone described as “a cool, American response to a ribbing.”

The positive response was partially driven by the fact that Parker and Stone aren't so much mocking Mormons as celebrating musicals. “We didn't sit down and say: ”All right, let's bash Mormons, how should we do it?“ Parker has remarked. ”We really just wanted to make a very traditional, classic musical.“

Thus there are dance scenes that echo the choreographic dazzle of Busby Berkeley and songs paying homage to Rodgers and Hammerstein, references to classic musicals including The Sound of Music, Bye Bye Birdie and The King and I (indeed its arguable the entire musical, with its fish-out-of-water missionaries learning about a new culture, is an updated homage to and send-up of The King and I) and winks to more modern fare such as Wicked.

“Trey and Matt know what they want and their brand of humour is very specific,” says Andrew Rannells, who played Elder Price on Broadway. “The formula is at heart very basic – it's a love story about these two guys who are friends – but they told it in a such a hilarious and new way that everyone was taken by surprise. Musicals can seem a little dated even at their best but this felt like a new idea, even though it couldn't have been more traditional in structure.”

It has a surprisingly wide appeal. When I reviewed the Broadway opening for this paper my only quibble concerned longevity: The Book of Mormon was packing in audiences curious to see a musical from the irreverent minds behind South Park, but how would ticket sales be affected once the celebrities left and The Book of Mormon was playing to coach-loads of tourists from middle America?

The answer was not at all. The Book of Mormon cost its backers $9.1m (£5.7m) and has gone on to gross more than $200m (£127m), allowing Parker and Stone to announce last month that they would be putting some of those profits into their own production studio, to be called Important Studios.

“I think those of us who had already seen the show in America love its irreverence,” says Cameron Mackintosh, who will rent The Prince of Wales Theatre to the show's producers for its run. “From the moment I heard about it I thought, 'wow this sounds wild and off the wall' – I went to see it in New York the minute it opened and loved it. What I really found remarkable about it was the huge cross-section of people in the crowd.”

Parker was more blunt about the show's widespread appeal, remarking caustically in a recent interview that: “When people say, 'will middle-aged people from middle America like this?', we're like, 'middle-aged people from middle America made this.”

So will the sell-out UK version be as rapturously received? According to Ambassador Theatre Group, one of the ticket agents for the show, it is expected to attract a younger, less traditional audience. “Our ticket-sales analysis shows a high rate of response from playgoers (as opposed to musical-theatre attendees) and a younger audience,” reveals Pat Westwell, the group's marketing director, adding that there was also evidence that discussions of the musical on social-media sites had driven pre-ticket sales.

“I think the practical thing is that they're not in any way trying to be cocky about it,” said Mackintosh, dismissing any possibility of a backlash. “They haven't taken anything for granted yet at the same time word of mouth has sold it out for months. London feels ready to embrace a genuine and original big musical hit.”

Encouragingly, Parker and Stone, who retain a hands-on involvement, are moving to London for the show's opening and, while Rannells and his co-star Josh Gad have swapped Broadway for Hollywood – Rannells is now star of sitcom New Normal and has a recurring role in Lena Dunham's Girls, Gad is the lead in the White House-set sitcom, 1600 Penn – the roles of Elders Price and Cunningham are played by the well-received national-tour cast of Gavin Creel, who has previously appeared in the West End in Hair and Mary Poppins, and Jared Gertner.

“I'll be interested to see how it translates for a British audience,” admits Brantley. “It's such a purely American show – I think of it as a happy, perverse Valentine to the American musical.”

He's right, but that doesn't mean that The Book of Mormon won't translate over here. Instead that sweet-and-sour combination of humour and heart should ensure Parker and Stone's musical wins converts on this side of the Atlantic as well.

'The Book of Mormon' begins previews at the Prince of Wales Theatre, London W1 (www.princeofwalestheatrelondon.info) on 25 February

This article will appear in tomorrow's print edition of Radar magazine