The Camerons actually live in the flat above No 11, but Downing Street is essentially one building. The moment David steps through the door of the flat into the working part of No 10, the place where he runs the country, he is Prime Minister. But this side of the door he is just Dad and is teased by his children about his cooking, his overly patriarchal interest in their school careers, his cackhanded attempts at DIY or abandoning them at the pub (more of that later).

Which is to say he is ribbed just like any other modern dad with three lively youngsters. They josh him about the Ikea kitchen cupboard he and Nick Clegg knocked together after a struggle, a symbol of the coalition perhaps.

A giant photograph of Ivan, their adored, disabled son who died aged six, dominates the wall opposite. Pictures of Ivan are on the mantelpiece and elsewhere.

‘The firm smack of government is needed,’ she says jokily. While the children dawdle at the breakfast table, Sam presses Dad to put down his top-secret papers and impose some fatherly discipline.

Nancy is telling Mum that the rehearsals for the forthcoming Trooping the Colour in Horse Guards Parade directly behind No 10 have kept her awake. 

But she is interrupted by Florence, who plonks herself on Dad’s lap on the sofa, taking precedence over any official papers.

The red box is cast aside as the Prime Minister brushes her hair, then grabs her clothes and pulls on her tights for her. 

She reaches for the hand of the most powerful man in Britain to take her to the bathroom. He then returns to ask Sam about the important question of the day: is or isn’t Florence allowed coloured hair clips at school?

‘Off you all go – you know what you need to do,’ says Sam, as the children each grab a toothbrush from a jar on the table and run to the bathroom. ‘You know what you need to do,’ echoes Dad. ‘Play!’ says Elwen cheekily. ‘No! Homework,’ barks Sam.

Elwen practises his sums and does spelling on the computer – until some nearby Lego bricks look more interesting. Dad guides him back to his spelling, gives a ‘kissee’ to Florence, plants a kiss on top of Elwen’s head and then one on the lips for Sam. He has a government to run; Sam has the school run.

As the car wends towards St Mary Abbots Church of England primary school in Kensington, Sam throws out words for Florence to spell: NOT, GO.

And then counting up to 30. For Elwen, it is spelling with ‘river’, ‘diver’ and ‘hover’. Florence worries if her apple is too big for her plastic box and if it has mysteriously started to smell. Don’t be silly, it’s delicious, insists Sam. No picky food fads allowed here. No crisps either.

Bright as a button, Nancy got her love of reading from David Walliams’s children’s books. ‘She loved his Gangsta Granny and Demon Dentist. He came to a Red Nose Day reception at No 10 and the children popped down when he was reading a story. They adore him,’ says Sam.

Then it is back to her three day jobs: working at Smythson, the smart Bond Street fashion brand (she was creative director until the election when she stood down to become ‘creative consultant’); then there is her weekly charity reception at No 10 to organise and, last but not least, her job of looking after ‘Dave’.Nancy is no slouch at sums either. Sitting in the car she runs through the six times table, forwards, backwards and randomly – and all correctly; the ‘three Rs’ rule in this family – and finally they arrive. Sam checks coats, homework and takes them in.

In her first full interview, Sam recalls the shock of moving into No 10 five years ago. ‘I remember day one, I was doing Nancy’s homework with her at home in W10 [their house in North Kensington] and suddenly Dave rings, 'You’d better get a dress on because we’re about to go and see the Queen.”

I thought, 'Oh my gosh, what am I going to wear?' I was five months pregnant at the time. We zoomed off to the palace – so surreal.’

After selecting a dress suitable for the Queen, she did one other fashion check and made sure the dolphin tattoo on her right foot was concealed beneath her tights, sparing the monarch a glimpse.

Sam casually whips off her shoe to show me the tattoo. ‘I had it done when I was travelling in Indonesia on my year off, just after my A-levels.’

In a tattoo parlour? ‘Yeah, it was just someone who we met and he did tattoos. It was really painful. I had it done on my foot because I thought, 'You’re going to have this for ever, so you might as well feel the pain.'

She points to it. ‘You can’t really see through my tights, but it’s just there.’ I can see the outline of a leaping dolphin. What do her children make of it? ‘Tattoos are everywhere now so I don’t think they've noticed.’

What if they come back from a gap year in their teens with a tattoo they’d had done in a remote tropical clime ‘from someone they just met’?

‘I’d be fine about it,’ she says without hesitation. 

‘Obviously, you’d have to explain that they might not want it in certain places where it would be for ever, but, no, I wouldn’t have any problem with them having a tattoo.’ 

I can’t help wondering whether her husband would be as broad-minded.

There is something more than slightly bohemian underneath her elegance and poise. The last album avid music fan Sam downloaded was by a US band called The War on Drugs and she is addicted to BBC 6 Music. ‘I listen to Radio 4 in the morning and the rest of the day I have 6 Music on. I love it.’

Another favourite is controversial psychedelic US band Poliça. Their latest album Shulamith features a grisly front cover with a woman’s back turned, skin bare, hair and neck matted in blood. The video for one track, ‘Tiff’, shows androgynous lead singer Channy Leaneagh waterboarding herself and smashing her own hands with a hammer. 

It is named in honour of Shulamith Firestone, a Canadian radical Marxist feminist who died in 2012 and was the author of The Dialectic of Sex: the Case for Feminist Revolution. 

Sam is no mere armchair listener but went to one of their gigs in Shoreditch. Recent books she has read include Mod by Richard Weight, about 1960s Mod culture, from fashion icon Twiggy to the Mods and Rockers riots in Brighton and Margate.

Her social media habits are just as eclectic. She shares videos, food, fashion and travel tips on Pinterest or Facebook; and uses the Shazam smartphone app which can identify even the most obscure piece of music from a few seconds’ riff.

Sam would be the first to admit that David, who got a first at Oxford, is more scholarly. She says she was ‘in the bottom set for maths – though the top one for English’. 

She went to St Helen & St Katharine school in Oxfordshire, Marlborough College for A-levels, then studied for a degree in fine art at university in Bristol.