I've always done it myself. It's not that difficult - you need to register for PAYE with HMRC as a small employer, and they have lots of guidance on their website. They have free software for which you essentially enter the pay details every time you pay and they calculate the net pay, employees NI, employers NI and PAYE tax for you. I used Excel to create a payslip template (based on mine!) and give that to the nanny which I just update every month using these numbers. Once a year you also need to submit annual returns, benefit statements etc. They send you reminders.
You need to give the information to the service provider in any case, and that for me was always the most complicated part (working out hours to pay, hours of holiday taken etc).
It did get complicated when I had to do maternity pay but otherwise it's relatively straightforward.