You can't (or at least you shouldn't) hire a nanny and tell them they are self-employed - it's against official guidelines, and the parents could get in trouble, plus they are clearly just tight bastards who don't want to pay the tax and NI.
Back in the day I knew a lot of nannies (from taking my DS when small to a playcentre), and I employed a nanny one day a week, who worked for a Famous Person (TM) the other four days. I lost a lot of respect for some of the middle-class parents I knew in this naice area after getting to know their nannies, and learning how much they paid them and how dismissively they treated them. As for my own nanny (who was brilliant, and should have been in her home country running some vast enterprise, not here having to wipe my son's arse), it took quite a lot to persuade her that when I paid her sick pay, or holiday pay, or paid for bank holidays, that I wasn't offering her charity money, but money she was entitled to. Same as tax and NI. She'd never had an employer pay for that before. I sometimes felt like the only person in London who was employing a nanny legally, despite being on a pretty low income myself.
Also £10/hour gross for a nanny in East Dulwich is laughable.
tl;dr the richer the parents, the tighter they are, and the more likely to treat the nanny like shit.