She sang at the inauguration of Barack Obama and at the diamond jubilee concert thrown for the Queen. She also performed at senator John McCain’s funeral and at Prince Charles’s 70th-birthday bash. Yet here’s Renée Fleming today, sitting in a dowdy London studio, eating salad from a cardboard box and feeling somewhat daunted.

“It is terrifying,” she says, of her part in the Tony-winning musical The Light in the Piazza. “There’s so much dialogue, which is not a skill I’ve practised much. But I’ve always had a voracious love of musical adventure.” Fortunately, her friend John Malkovich has given her some advice. “He told me, ‘You just have to put in the hours.’ That made me feel better.”

Fleming’s confession comes as a surprise, given that she is a star who has been at the very top for more than three decades, with a repertoire that runs to more than 50 of opera’s great soprano roles. The Light in the Piazza, composed by Adam Guettel, is a bit of a departure for her, though: not only does it mix musical genres, but it throws her in with quite a varied bunch. Her daughter, Clara, is played by 23-year-old Dove Cameron, best known for playing both title characters in the Disney Channel’s twins comedy Liv and Maddie.

When should a mother let go of her daughter. That’s the dilemma that faces, Margaret, when Clara falls in love during a sight-seeing tour of Italy. Her predicament is laid out in the opening song of the musical, which is being staged at London’s Royal Festival Hall.

Fleming, who is 60 and based in New York, has two adult daughters of her own, but this is the first time she has played a mother on stage. It is the fate of the lyric soprano whose voice doesn’t age to be locked into younger parts and romantic repertoire. But two years ago, she called time on one of her signature roles: the Marschallin, an aristocrat who takes a young lover in Strauss’s Der Rosenkavalier. “Most of my roles were ingenues and I said, enough is enough – it’s not really working any more. It was time to give someone new a chance.”

Far from putting the brakes on her career, this proved a breakout moment. The two years since have been full of theatre, with an eight-month Broadway run of Carousel followed by an edgy off-Broadway show, Norma Jeane Baker of Troy. In the latter, Fleming played a speccy stenographer hired by businessman Ben Whishaw, whose efforts to understand the iconography of Marilyn Monroe – via Greek myth – lead him to metamorphose into her. Not only did Fleming appear in person, but her sampled disembodied voice became the soundscape of the show.

It is not only on stage that Fleming’s voice has developed a life of its own: there can’t be many movie-goers who haven’t heard it, though they may not realise. Fleming did the vocals for Twilight and Shadow, a song from The Lord of the Rings written in Sindarin, the tongue invented by JRR Tolkien to be spoken by the Immortal Elves of his Middle Earth.

More recently, she was the voice of Bianca Castafiore, the Captain Haddock-bothering opera singer in Steven Spielberg’s The Adventures of Tintin – her top C electronically ratcheted up to a glass-breaking top F. Last year alone, her singing featured on Guillermo del Toro’s Oscar-winning The Shape of Water and – unbeknown even to her – contributed to Three Billboards Outside Ebbing, Missouri. Its soundtrack includes her recording of Last Rose of Summer, from the opera Martha. “They licensed it,” she says. “I didn’t know until I saw it in a review.”

What is it that keeps her seeking out new musical adventures? “Brave or crazy, I don’t know which, but I work a lot,” she says, attributing her work ethic to her Czech ancestry. Both her mother and father were music teachers. “I wanted to be the first lady president or a veterinarian.”

On BBC Radio Four’s Desert Island Discs in 2005, she chose her own performance of Joni Mitchell’s River. “I was probably 16 or 17 when I heard Joni Mitchell and thought, ‘That’s my voice.’ I played the guitar and went through a whole singer-songwriter era.” It was a rebellion of sorts for a “good girl who wanted to be bad, to wear nylons and smoke in the bathroom”. At college, she sang with a jazz trio, only opting decisively for a classical career after being forced to choose between dropping out to tour with the Illinois Jacquet Big Band or going on to graduate school.

Fleming’s career has been so dazzling, it is a surprise to learn how slow it was to take off. In her early 20s, she failed a big audition but now regards it as a blessing. “My voice wasn’t worked out, I had a lot of technical flaws.” She took herself off for classes with legendary soprano Elisabeth Schwarzkopf, who piled shame on her: “You were either ‘golden’ or you shouldn’t be allowed to sing at all.” One of several bouts of stage fright followed. How has that experience informed the masterclasses that she now gives to young singers? “I don’t use the singers to make the audience laugh,” she says, pointedly.

By her own account, her biggest break came in her mid-30s, through an understudy role. The Desdemona in a top-flight production of Verdi’s Otello hurt her back, plunging Fleming into the part – and a confrontation scene with a giant she had never even met: Plácido Domingo. “He was so frightening that, by the end, I needed help to get off the stage,” she says. “And when it was over, he held out his hand and said, ‘Hello, I’m Plácido Domingo.’ It’s very important to be prepared. That Otello put me on the map.” She continued to sing Desdemona up to 2012.

Perhaps her most life-changing break happened off stage, though, after she was introduced to author Ann Patchett by someone who mistakenly believed that Fleming was the inspiration for Bel Canto, Patchett’s Orange prize-winning novel about an opera singer taken hostage. The two women became close friends. And Patchett introduced her to a lawyer friend of hers, Tim Jessell, who in 2011 became Fleming’s second husband. Fleming went on to provide Julianne Moore’s singing voice in last year’s film adaptation of Bel Canto.

“I love finding interesting ways of making classical singing accessible to the rest of the world,” she says, “because the opera world is increasingly small. We’ve been presenting it in the same way for so long that it’s ceased to be relevant.” Which is why she is so keen to venture beyond the classical repertoire. Between performances, recordings and masterclasses, she works with Washington’s Kennedy Center, bringing neurologists and artists together to promote the use of music for people with brain injuries.

In 2009, after a guest appearance on Elvis Costello’s TV show, Spectacle, she was signed on for an album, Dark Hope, covering songs by Leonard Cohen, Jefferson Airplane and Tears for Fears. It’s a mark of the importance of family to Fleming that her backing singers on Dark Hope included her cabaret-singer sister, Rachelle, and her own daughters, Amelia and Sage. They are now 26 and 28 and, although both were promising singers, they showed no sign of allowing themselves to be “indentured” into a musical career. “It’s a hard lifestyle,” says Fleming with a laugh, “and they know too much.”

• The Light in the Piazza is at Royal Festival Hall, London, until 5 July.