Yes you should register as an employer and produce payslips, deducting tax/ni if appropriate to do so.
Have you spoken with any of the nanny payroll companies and explained that the nanny has other income and thus all their income in this employment will be taxable?
On the plus side, there is no Employers NI to pay at that salary level.
HMRC says: Source
If you're employed on a casual, temporary or part-time basis, by law your employer must:
deduct tax and National Insurance contributions from your wages
give you payslips
deduct student loan repayments, if relevant
give you form P45 when you leave
give you form P60 at the end of each tax year if you still work for the employer
Also, we would like it to be term-time only, which all the nannies we have interviewed have said is fine, however i would like to give them some kind of holiday pay otherwise it feels a bit exploitative.
Term time workers get holiday at 12.07% of their working time.
So if someone is doing 6 hours work per week, 38 weeks a year:
6 x 38 = 228 hours
228 / 100 = 2.28
2.28 * 12.07 = 27.519
So holiday is minimum of 27.52 hours, which you can round up but not down. So 28 hours?
That could then be split across the terms, so 9 hours for term 1 and 2, 10 hours for term 3.
This I believe can be paid along with the salary for the term / month in which it occurs. They are sort of working a few days during some of the school holiday.
Some people may want salary split over 12 months, this is also possible but can create a problem if they leave the job, or the job ends, before end of the year (year starting from the employment start date).
You would need to check a calendar, as schools do different term lengths. Also check if the working day is ever on a bank/public holiday - there are not many that ever occur during school term time but they can occur, especially if the nannies working day is a Monday.