We have this arrangement, with a nanny doing her own studies. Ooh, this is going to be one of my (even) longer posts.
The working hours arrangement is: year-round she has a specified number of hours, ie M-F, x am to y pm; but in term-time a specified no. of those hours weekly are - subject to childcare emergencies, like kids sick off school, this is v v important to state in contract - hers free to devote to studies, whatever else she wants to do etc. And because she has this specified no. of free hours, the pay is reduced a bit (not very much - by about £2 per free hour) from what we pay for full-time. (Our current nanny is in no conceivable way an advantage-taker, but doing it this way - ie, these are your notional hours but you get this much off - would give you a comeback if you had a term-time nanny who was really starting to cut corners on the time she devotes to the job.)
Because she does more term-time weeks than full-time, the pay our nanny's registered with at the payroll people (we use Nannytax) is the term-time rate. We spoke to Nannytax, and all they need is for us to tell us how many full-time weeks she does per quarter, and they'll do the totals for tax & NI taking that into account. Nannytax weren't phased by it at all.
Holidays: well, since ours is year-round she gets the standard of four weeks paid - I wouldn't discount that for any part-time considerations. Not sure about someone just doing a short (summer) contract - I guess there may be legal guidelines about at what length of contract the right to paid leave kicks in. Try ACAS?
We are definitely her employer, couldn't say otherwise. But I do know nannies who do short-term emergency nannying who have been able to work on a self-employed basis. Again, I don't know what sort of duration their emergency contracts are. Not sure who to take that question to - Nannytax have a bit of a vested interest in saying you've got to pay tax, I think! However, agreeing a gross salary sounds in those circumstances like a very sensible one. If she's uncertain about doing that, you could perhaps take out a subscription to one of the payroll companies so's to give her access to helplines?