Everyone I ask about David Babani reaches within seconds for the word “chutzpah”. This conjures an image of an audacious, gabby, can-do, darting sort of a chap. This, however, runs wildly contradictory to the big, baggy-eyed, exhausted-looking bloke, wearing a hanging-out crumpled shirt, who limps across the restaurant of the Menier Chocolate Factory to greet me. To be fair to Babani, he has just flown in from New York. But today, at least, he looks older than his 28 years. Indeed, with his burly, sloping-shouldered physique and lumbering lope, he resembles a dancing bear with a dodgy ankle.

Babani, who turns 29 on Saturday, is of an age when some have barely started a career. Yet already he has established himself as a force to be reckoned with on the capital’s cultural scene. Only three years ago, he and the restaurateur Danielle Tarento took over the lease of the Menier, in Southwark, a characterful multi-storey warehouse, built as a factory by a 19th-century chocolatier, Emile Menier. Its arts-loving landlord had roughly converted it into a gallery and performance venue, but to be commercially viable, it needed someone to programme the space effectively.

That is where Babani’s experience came in: since he and Tarento took over, they have had several critically lauded sellout shows and two West End transfers. Last Christmas’s inspired revival of Stephen Sondheim’s Sunday in the Park with George became one of London’s hottest tickets. After an extended run at its 180-seat home, the luminous production transferred to Wyndham’s theatre, in the West End, where, despite offering no huge stars to distract from sultry temperatures, it ran until September. It even won Sondheim’s approval. “To be seated at the back next to Stephen Sondheim on a wet wintry Wednesday afternoon and watch him being moved to tears by our production,” Babani says, shaking his head at the memory. “Well, you could retire now.” Babani’s recent Stateside trip was in part to discuss a Broadway run. “If it happens, then we’ll recoup our spend and it will go into profit,” he adds.

Babani is too wary to jinx himself by saying it outright, but he clearly hopes for a similar success with this year’s Christmas musical, Little Shop of Horrors. The fond send-up of 1960s sci-fi, set in a Skid Row florist where one plant has a mind of its own, is a canny choice. Alan Menken and Howard Ashman’s 1982 off-Broadway hit is a show lots of us think we know well when in fact we don’t. In America, it is overexposed; performed in high schools every week. In London, there has not been a production in 20 years. The film was made in 1986.

One of Babani’s coups has been to persuade the writer’s estate to agree to a remodelling of the plant using modern animatronics. The puppeteer Andy Heath will operate the plant, Audrey II, which will be voiced backstage by Whose Line Is It Anyway’s Mike McShane. The on-stage cast will include Paul Keating, Sheridan Smith and Jasper Britton, all experienced theatre or television actors. The cast of 10 will be joined by a live band. As we speak in the bar, they are all in a sun-filled upstairs rehearsal room, being put through their paces by the director Matthew White and the show’s choreographer, Lynne Page.

The choice of one of off- Broadway’s biggest-ever hits fits in neatly with the Chocolate Factory’s programming ethos. It could be seen as a very off-Broadway venue. Babani’s role as an artistic director who produces rather than directs follows an American model. Also, his biggest successes have been American. This is partly pragmatic. “There are better- established new-writing theatres, like the Royal Court,” he says. “So we’ve cultivated a path where we put on new American writing.”

These plays may be serious, but the primary purpose is to entertain. Customer loyalty is promoted through providing a competitively priced night out. Most opt for the meal deal, which offers a ticket and two courses for less than £30. Since the Menier has no public funding, its future relies on revenue from ticket sales and the restaurant. “It would be good to get some sponsorship, but you’re freer without public funding,” says Babani. “I don’t want to be made to put on lesbian dwarf theatre from Kazakhstan.”

So far, his programming has built up a loyal audience, many of whom travel from the leafy enclaves of north London, where Babani, whose father is a former publisher of Sephardic Jewish origins and whose mother is “a very English” professional bridge player, was brought up.

His producing career began at Highgate school. In the summer holidays at the end of his first year at Bristol University, he produced Sondheim’s Assassins at the New End theatre, Hampstead, with a professional cast. Sondheim’s people were so impressed, they invited him to New York to justify his changes to the great man. Wasn’t he, at 20, cowed? “No,” he says, apparently surprised by the notion. “He asked very direct questions, so I just had to hold my ground.” Shortly afterwards, he was offered the stewardship of the Jermyn Street theatre and took indefinite leave from university.

Having got the restaurant up and running, Tarento is leaving the Menier to pursue other projects. This means that if Little Shop of Horrors — at £300,000, the theatre’s most expensive project to date — fails, it will all be down to Babani. “We can’t really afford a flop,” he admits. “But it wouldn’t sink us. We’d just have to put on some really cheap shows next year.

Film screenings, perhaps?” he says, grinning.

Little Shop of Horrors is at the Menier Chocolate Factory, London SE1, Nov 17-Feb 25 (020-7907 7060)