we are thinking of employing a nanny for the first time, 2 days pw.
You may therefore not be their main employer and thus may need to operate PAYE using Basic Rate tax. The nanny may decide to request the tax office splits their tax code between jobs, in which case you then operate PAYE using the taxcode provided to you by HMRC.
Probably best if you have a read of PAYE for Employers. P49 (2.79MB PDF) is a good thing to read as a starter as it talks you through the first payroll run.
When you register with HMRC you will get a New Employer Pack which includes a CD-ROM. That CD includes a P11 calculator to help you with calculating and recording the Tax and NICs amounts.
One thing I don't think it will do though is to produce a payslip. So you will need a way of doing that... could be done using a spreadsheet... or manually. A past employer of mine has done it manually using a carbon copy-pad, thus producing one copy for me and one copy for the file.
Also, is it reasonable to ask the nanny to take her holidays during the school hols?
Yes, you can dictate when holidays are taken. Be upfront with applicants for the job so it does not come as a surprise to them when they see the contract.
And how about asking her to take all bank hols as holidays?
Legally that is fine. It may be worth you working out the specific dates affected over the next few years. UK Bank Holidays
We don't have much money, and can't really afford to pay for a nanny when we're not working.
Nannies are contracted to work in the same way as other employees so have the same employment rights. So if you didn't work on a day that your nanny would be working... then if you didn't want the nanny to come to work you would still need to pay them. So if you take holiday, probably best to take it when the nanny takes their holiday.
Consider what happens over periods such as Christmas and Easter where some companies may shut-down for a while. If your employer is shut but your nanny isn't on holiday, then you will still need to pay your nanny - though of course you could ask them come into work.