The guides on this site are helpful summaries, although the guidance is necessarily general.
What sort of hourly rate would I pay (London) This depends where in London (zone 1 is more expensive than zone 6, west is often more expensive than east) and how qualified/experienced the nanny is. £10gross/hour minimum, £13gross is more realistic, top candidates will be £15/16gross
What do I need to provide at home - food etc? (it would be live out) Food, heating, lighting etc but nanny should eat roughly what your DS does. A small kitty for activities/emergency bread and milk. Not quite at home but you will need to consider whether you pay for a car/oyster card or whether everything is walkable.
I understand I need to be their employer - where cani find info on tax, NI, holiday and sickness pay? A payroll company is probably a good summary. Although HMRC will have the best info on tax, it's not always the most accessibly written! Nannytax are expensive but have a pretty comprehensive website. As your nanny will be PT remember to negotiate a gross wage. Holiday is 5.6 weeks, pro rata for a PT employee.
Do they have to take their holiday when we take ours? Generally it's Bank Holidays off, leaving 4 weeks of which 2 get chosen by the nanny and 2 by you. If you take more holiday then you will still need to pay your nanny although some people have their nanny batch cook or dirt through toys.
What a the chances of finding a nanny prepared to work part time (hoping for a p/t return) Quite good! But it does depend on your days.
Are agencies a good option or an expensive sifting process? I personally think they're an expensive sifting process but it depends how confident you are about what you're looking for in a nanny and how well you can stick to your guns so you can narrow your pool of candidates according to your initial requirements.
What have I forgotten to consider?! Have you thought about the possibility of a nanny share? Would you accept a nanny with their own child? Would you want to claim childcare vouchers, thus needing an OFSTED reg nanny? What kind of lead in/settling time do you want to have?