I think it is rather complex, so not sure if I will get this right.
Holidays - you can dictate when all holidays occur, so that isn't a problem. However you may end up wanting them to not work on more occasions than they have statutory entitlement to take off (see below about how many days entitlement they might get, plus the number of bank/public holidays).
How many weeks (and thus hours) would your nanny be working a year?
Lets say a nanny works 10 hours a day, 5 days a week thus 50 hours a week and works 16 weeks a year.
16 weeks x 50 hours = 800 hours
800 hours x 12.07% = 96.56 hours
96.56 hours / 10 = 9.66 days
Round that up to give 10 days.
The 12.07% is the leave entitlement that accrues as a worker works. See Casual Workers
With regard to Bank Holidays quite a few do occur during school holidays, such as Boxing Day, New Years Day, Good Friday, Easter Monday, depending on when half term falls Spring Bank Holiday and Diamond Jubilee, Summer Bank Holiday. See: Bank Holidays
Do you need them to work any of these dates? Do you want to insist that they take them as part of annual leave (could well use up most of their entitlement), or are you in a position to let them have the day off on full pay?
Also if I include the bank holidays in the allowance does that mean if there are more bank hols than she is entitled to during the holidays then she has to use her remaining holiday allowance to cover it?
I don't get what you mean by that. If you include the bank holidays in the allowance then I think they may only have 3 or perhaps 4 days remaining. If you then go away during the Summer Holiday, then I would guess that you would go away for those remaining 4 days, though probably for longer.
If you choose to not need them on days which are during the school holiday periods, then I feel that if they don't have sufficient holiday that can be taken you will need to pay them as if they were working. You are making the decision to not need them, not them choosing to take holiday.
I will pay her tax and NI for the weeks she works but will her employment still be considered to be continuous? Would it be better to average it out as a monthly salary? Would that work out better for her?
No idea on that one I'm afraid. You should get advise about if it is continuous or not, plus also about taxation side of things.
What if your nanny wants to leave... if you are paying them a set amount each month and they leave before the Summer holidays, then they may well have been overpaid, as the bulk of the work is during the summer.