When Hannah Rothschild was growing up amid fabled wealth and unimaginable privilege, her father would "drag his children around museums" despite the fact that at home they were surrounded by perhaps more masterpieces than you could shake a paint brush at.

Always, though, she recalls longing for paintings to speak.

"I remember thinking, 'Why aren't these things talking to me?' They seemed so inanimate, so I always had this fantasy going. There's a painting in the National Gallery of Gainsborough's daughters, chasing a butterfly. One sister is dragging the other and one of my sisters was exactly the same age. I thought, 'Are they fighting? Or is it like me dragging my sister around?' I recall wondering, 'What happened to them? What will happen to us?'

"I have a theory that paintings accrue more beauty over the centuries, as well as dirt! I always ask, 'What have they seen?'" says the auburn-haired, honey-voiced writer and documentary-maker, who celebrates her 53rd birthday days after the publication of her accomplished debut novel. In The Improbability Of Love, the improbability of a work of art adopting a mischievous narrative voice is wholly credible.

Her pacy satire of the art world - soon Rothschild will be the first woman to chair the board of National Gallery - centres on a lost (fictional) masterpiece by French Rococo painter Jean-Antoine Watteau; its history is "strewn with sex and love and lust and even a dead body or two".

Found in a junk shop by lonely Annie, who works as a private cook, the painting is bought on a whim for her lover. It owns a terrible secret that someone is prepared to kill for. Rothschild dishes up a salmagundi of unscrupulous dealers, desperate auctioneers and dodgy art experts, with a side-order of scheming Russian oligarchs. It's on the money and as delectable as Annie's bizarre recipe for "pigeons dressed like baby peacocks, feathers made from herbs, captured in cages of spun sugar".

Educated at St Paul's, Marlborough and Oxford, although she dropped out before gaining a history degree, Rothschild has written a novel that is so pleasurable I've read it twice, and will read it again. When I tell her this, her eyes fill with tears. She dashes them away, apologising as if she had never received a compliment before, although her book is being translated into six languages and there's a battle for film rights.

I am not alone in my admiration: the writer Elizabeth Gilbert enthuses: "It is a romp, a joy, and an inspired feast of clever delights," while novelist Barbara Trapido writes: "It's funny, sad, profound... It's my Book of the Year."

In its beguiling voice, Watteau's painting declares: "Value accrues by association... If I tell you I have been owned by kings, queens, a Holy Roman emperor, a pope, a great philosopher and a few others, you become interested. As the decades rolled by, as I was passed from one illustrious owner to another, my value increased. Who wouldn't want to be linked to past glory, to monumental power?"

Despite Rothschild's air of self-effacement, she knows a great deal about illustrious owners, past glories and monumental power, fabulous treasure-filled houses and the shark-infested art world. She is, after all, a member of the multinational banking Rothschild dynasty, with its penchant for collecting expensive art. She was on the horns of a dilemma as to whether to centre her plot on contemporary art or old masters. "I see both sides as I'm involved with the Tate, as well as the National Gallery. I have one character remark: 'Old masters are all warm white wine and nibbles, blue stockings and terrible ankles. Modern art is martinis and sushi.' It really is like that and, yes, Damien Hirst is indeed 'the David Beckham of the art world,' as I write."

She chose Watteau because as a young woman, living unhappily in Paris, she found solace in his enchanting, melancholic 1712 painting, Pierrot. Also, little is known about Watteau's penurious existence - he died young. "Lucian Freud was obsessed by Watteau," she reveals. "I heard him talking about him - this artist, who painted great slabs of flesh loved the light, airy Watteau!"

We meet in her father, Lord Rothschild's offices, in St James's, London, where posh art galleries are outnumbered only by gentlemen's clubs. It's secreted behind a discreet door. You enter a hallway whispering wealth. Expensive art and sculptures are everywhere: a Sickert and a Giacometti on the sweeping staircase alone, according to one journalist who preceded me into these hallowed halls.

After a distinguished career that has taken her from the BBC, where she made several films about auction houses and lost artworks, to scripting movies for Ridley Scott, writing for glossy magazines and making documentary features, Rothschild now works with her 78-year-old father, Jacob, in the family business four days a week. She spends another day at neo-Renaissance Waddesdon Manor, near Aylesbury, bequeathed by the Rothschilds to the National Trust in 1957. It is still administered by a trust under her father's chairmanship. His eldest child, Rothschild, is also a trustee, as well as a trustee at the Tate and vice-president of the Hay Literary Festival. She takes the National Gallery chair in August, but has been heavily involved in the selection process to find a new director: Luke Syson from New York's Metropolitan Museum of Art.

"It's all quite grown-up, isn't it?" she says.

So, given that she has a day job in high finance and three daughters - she and their American father, William Lord Brookfield divorced after six years of marriage - heaven only knows how she found the time, or the energy, to write The Improbability Of Love, her second book. Her first, The Baroness, published in 2012, is a biography of her aristocratic great-aunt Pannonica, or Nica, who escaped from the gilded cage of her life and bolted to the US, where she devoted herself to American jazz musician Thelonius Monk.

It's a terrific book and the subject of one of Rothschild's films. As a fly-on-the-wall documentary-maker, she's also turned her enquiring eyes on Peter Mandleson - "He loved it! There's no accounting for taste is there? But politicians, like paintings, do love being looked at" - Joan Collins, Frank Auerbach and Nicky Haslam, professional socialite and interior designer to aristos and rock stars, who appears in The Improbability Of Love as waspish Barty, although Rothschild insists that no other characters are based on real people.

"I sent it to Nicky and he sent me a note saying, 'Bone china, very common. Only nannies drink tea out of bone china.' He's so funny. I find him life-enhancing, one of the great characters of our time."

She has, though, drawn on real events from a dark period in her own family history during the Second World War, when the Nazis plundered tens of thousands of works of art from European Jews, such as the Rothschilds - "the Nazi party used my family as the model for the apotheosis of evil Judaism". More than 5,000 works were looted from the French branch alone. Many were returned, other precious objects and pictures are still missing. "One lost, Dutch painting turned up in a vault in Switzerland very recently," she says.

After all the fact-checking that went into The Baroness, she thought that she would never have to worry about facts in fiction. "Of course, I bit off more than... To go from the 1700s to 2015 and beyond, as I do in the novel, I soon realised that fiction only works if rooted in fact. Authenticity is important. In both books, I had to resist the temptation to put down everything I knew; it's knowing what to leave out."

Which was no small task since she's been working on the novel "for decades, which is really pathetic". It might still have languished in its first draft in a drawer had her publishers not urged her to produce a second book, following the runaway success of The Baroness. "But I'm writing about a world I know - I have met so many of these people: Russian oligarchs, for instance. Some are monsters, some are not. I have an idea for another book - but have no time to write it."

Did she cook any of the dishes that Annie serves up at two lavish dinner parties in the novel? Her "piece de resistance", a boneless turkey stuffed with a boneless goose, stuffed with a boneless chicken, a boneless partridge, a quail and finally a baby snipe, perhaps?

Rothschild responds with gales of laughter. "I cooked everything in the book, so I suppose I've become a method writer. I'm not a good cook, but I love eating... Look!" she exclaims, indicating her hips. "Many artists are great cooks, though. Lucian Freud would cook the most delicious woodcock, and Marc Quinn once cooked that stuffed turkey thing for a New Year's Eve supper. It had to be carved with a miniature saw, but tasted incredible. I managed only to get a chicken inside the turkey... the kitchen was a disaster zone."

The Improbability Of Love by Hannah Rothschild is published by Bloomsbury, priced £14.99