I would wonder if a childminder might be an option to consider, as you are not wanting childcare that late into the evening.
Would we find anyone to work this small number of hours? (in the home counties).
You won't know until you try.
What is the tax position, would we be the employer?
I feel you would be the employer as you are dictating the hours, duties, expect them to turn up and not send someone in their place, regular fixed hours not ad-hoc. You are wanting an employer:employee relationship I expect, in that you want to be able to tell them what to do, when to do it.
Unlikely to be the person's only job, so you may be the secondary employer, or one of many employers.
There would be no Employers NI due as the pay is below the Lower Earnings Limit for NICs. Limit is £109 a week (2013/14 tax year).
If they are on BR taxcode, then if paying £60 a week, there is £12 per week income tax (2013/14 tax year).
Perhaps call a nanny payroll company and see if they can offer any free advice.
What would the going rate be? I was thinking £10 an hour, is that enough?
Probably enough to get some interest, though best to say that's a Gross salary, so before any tax deductions.
What would my position be if I wanted to change things in the holidays (e.g. 6 hours on one day) if it fitted the persons schedule?
You would need to agree that with the individual concerned, in advance. It would increase the pay, so increase the income tax and could mean you get to a point where you pay Employers NI and deduct employee NI
It will add complexity to statutory holiday entitlement. 5.6 weeks x 6 hours is an easy calculation (33.6 hours annual holiday entitlement).
Doing the occasional few hours extra may be overtime but having it as more regular I feel would mean that holiday entitlement should accrue. So you may need to use the 12.07% of hours worked calculation.
ACAS: Holidays and Holiday Pay (pdf)
The DCs dad works from home in a separate office. Other than a brief hello the children don't bother him before six. Would this be a problem for a nanny?
Shouldn't be as long as he stays in the office and it is sufficiently sound proofed so the children do not have to be quiet all the time.
Is there anything I haven't thought of?
Transportation (or would they always walk, regardless of circumstances)
School closure/teacher training days - more hours needed then?
Timings - how long is the walk from home to school? Nanny in my view starts work from your house, not from the school gate.
Parking outside your home... if nanny drives, where do they park their car?
How local do you want the nanny to live? Given the low number of hours, it may not be very cost effective for someone to be travelling very far to work. Whilst they may feel it's worth travelling 50 miles, do you? Consider what you feel is reasonable and restrict your hunt for applicants to that mileage radius (plus a bit maybe). Recruitment can take a while, so you don't want someone to start the job and then leave after a month or so once they realise how much it is costing them to get to/from work.
The two afternoons - I am presuming you would choose when those are and that they would not change... is that the case?
Just my view. Definitely try to get some professional advice about the tax side of things.