My employers have used NannyPaye and PayeForNannies. I have had no issues with either, and the parents did not report any issues either. So either of those are fine. They will be a very similar cost.
Opt for the pension scheme management if it is an optional extra. They can often integrate their payroll system to some of the pension providers, typically using NEST Pensions.
Yes, 3 days per week will attract some people and may put others off. Make it clear if the position is term time only or all year round. If you intend to limit when holiday can be taken, make that clear, as some nannies like to take holiday in term time as it is often lower cost.
I would start advertising from September, and increase ads in December. Sometimes people know when they are likely to be available but plans change. So get a feel for who is a likely candidate so you can make a decision in January/early Feb.
Setting up payroll can take a few weeks, so if first pay is end of March I would start that process end of February. Contact payroll for an initial chat before then, so you have some cost calculations so know what gross salary you can offer.
Ofsted - in England. Tax Free Childcare scheme can be used with a registered nanny. If you need someone who is registered, make that clear in adverts and see their registration certificate at interview. The registration process can take many months, so I would avoid recruiting someone who is not already registered.
You CANNOT use funded education hours with a nanny. Use that for the pre-school.
Do your cost calculations carefully.
Think about how your nanny will transport your children, if they need their car then you will need to cover costs, typically 45p per mile (maintenance, insurance, wear & tear, the fuel itself) and provide somewhere they can park (if in a parking controlled zone it can be very hard to get a permit for a nanny from some councils).