"Hi, I'm Mary," she says quietly as she creeps self-consciously into the arty media hangout in London's Notting Hill where we've arranged to meet. She's easy not to notice among the other better-known faces that surround us; slim and diminutive, she wears no make-up and is dressed down in black Marc Jacobs blouse, slimline grey hipsters (by her sister Stella, of course) and understated flat pumps. It's almost as if fame is anathema to the 42-year-old whose image was first etched on the public consciousness as a baby cradled in her rock star father's jacket which featured on the cover of his 1970 album McCartney. The picture was taken by her mother, the photographer, vegetarian and animal rights campaigner Linda Eastman McCartney, who died from breast cancer in 1998.
As a successful photographer in her own right, Mary is on first-name terms with the likes of actors and models such as Ralph Fiennes, Jude Law, Lily Cole, Lizzie Jagger and Kate Moss.
Yet apart from her eyes, which resemble her father's, the only clue to her celebrity status is the number of times she's stopped on her way to our table by people desperate to say hello. She deals with these interruptions politely but with a detachment that screams (albeit quietly): let me be.
Even her food is as unfussy as it's possible to be. Her first cook book, simply entitled Food, is aimed at demystifying meat-free home cooking (like her siblings Stella and James, and older half-sister Heather, Mary's been a vegetarian all her life) and while attractively presented, her recipes are disarmingly simple to the point of basic. The photography, also by her, is naturalistic and homely rather than expensively styled.
"I just photographed the food as it was," she says. "Some of it is half-eaten, like my husband's pan-fried cheese, tomato and onion sandwich, which I borrowed while he was in the middle of eating it. I like seeing parts of a meal on the table. Other dishes, such as spaghetti with tomato and red wine sauce, I placed on the kitchen windowsill to give it as much natural light as possible. I wanted it to look like the genuine at-home food it actually is."
What challenges did she encounter with food photography, a notoriously difficult genre? "As food is quite static, getting it to look tasty and vibrant was difficult at times. Tiny details, such as whether a fork is placed straight or crooked, make a huge impact and deciding how you want it to look can take time. Another challenge was not to over-complicate it. I really wanted to communicate the message that it's possible to have tasty food that doesn't take a lot of time to prepare and make, and which doesn't require a larder full of fancy incredients."
The book is aimed not only at meat-avoiders but also at the growing number of meat-reducers – a demographic that is helping grow the popularity of the Linda McCartney Foods range.
Over a pot of green tea – she declines to eat, as she is due at a tasting of new products for the brand founded by her mother, for which she is a development consultant – we discuss the importance of a mother's influence over her children's eating and health, and how her own early eating experiences at her parents' cottage in the Mull of Kintyre have had a profound lifelong effect.
"I got into cooking through watching my mum cook, hanging with her and chatting," she says. "She liked to have company in our big kitchen-diner, so we'd all be with her as she prepped vegetables for our meals. Often the radio would be on. Mum was a very intuitive, instinctive cook and I learned to make things like stews, sandwiches, salads and wraps from her. You pick it up and carry it on.
"In my own house we don't have a separate dining area. Our dining table is in the kitchen, so everyone can sit around. That social aspect of cooking and eating is very important. My boys all love cooking and take an active role in choosing the amount of seasoning and which herbs to put in the tomato sauce, for example. Recently my oldest suggested adding red peppers to tomato sauce and that's good. Likewise, if you chop veg into pieces that are too big they won't eat it, so I ask them what size they want. Children are more likely to eat food if they've had an involvement in its preparation.
"I like to think the old days of having a meal presented as a finished product in a separate dining room, or living room, without seeing how it was made, are over."
It was through her French-American step-grandmother that Mary learned the art of pastry cooking and the discipline of writing down recipes, and a gift of a wooden recipe box from her Aunt Louise encouraged her to start keeping records of what she liked to cook.
Linda's creamy tomato soup, one of many of her mother's recipes which features in the book, means the most to Mary. "Mum loved making it. I remember having it for lunch in the summer after a long walk in the Mull of Kintyre. It's still a favourite for me and even the smell can return me there."
She may have learned cooking through the female line of her family but she herself has four boys, the youngest barely eight months old, and they don't have their maternal grandmother to learn from directly. I ask if there's anything her boys have learned to cook from their famous grandfather, who is also a committed vegetarian. She says: "Dad does a mean barbecue and he also makes lasagne when we visit him [at the family home in Sussex]. Oh, and he makes a great bagel with dill pickle, avocado, humous, baby spinach. It's yummy, really good, and I could eat it right now. When he comes to us, I cook for him."
Mary's two older sons from her first marriage to television producer Alistair Donald are Arthur, 13 (Paul's first grandchild) and Eliot, 9. Her two youngest, Sam, 3, and baby Sid are with her second husband, the director Simon Aboud. Does she think it's perhaps more important for boys to learn about food than girls?
"I think it's not just good for boys, it's good for all children to know what goes behind making something," she replies, adding with a smile, "although I do notice that my boys are fascinated by meringues. They're interested to see how egg whites whip up; they like to know the science behind it."
Does she worry that her sons will lose their interest in food as they grow up?
"Even if they do I am sure they will come back to it," she says. "When I left home there were certain recipes Mum used to make that I started, only to find they didn't taste the same. So I'd ask her what I was doing wrong and she'd give me tips. You can come back to and adapt recipes you've learned early on. I hope that will happen with my boys.
"I can remember when we were children on holiday at the cottage in Scotland, my parents telling us, 'This is the most amazing turnip, you know, because we grew it right here!' and 'This is the most wonderful vegetable stew, because it is super-fresh' and we'd be rolling our eyes and saying, like, 'Boring!'
"And in the mornings I'd sneak down to eat the fresh peas in their pods, thinking I was being very naughty, only to discover my parents were absolutely delighted.
"It was only much later that I could understand why they were so enthusiasatic. There's nothing like growing and eating your own, and they were growing potatoes, carrots, turnips, beetroots. It was quite avant garde in a way.
"It was in Scotland that I first learned where food came from. Having that experience of the countryside also raised my awareness of our impact on the environment. I have my parents to thank for that."
Food by Mary McCartney is published by Chatto & Windus, priced £20.