On the question of HMRC requirements and inputs you should read this webpage
and also this one.

If you haven't already done so I would suggest you contact HMRC (Employers Helpline) to find out if you need to register as an employer and if you need to have a PAYE Employer Scheme set up in your name with your nanny as the employee.
If so there will be a small amount of work involved with this, but as there is likely to be minimal tax or NI deductions it should be an easy input from your side which can be done online.

It looks like you're already paying wages above the National Insurance lower earnings limit, so a PAYE scheme may well be needed.
It's been quite some time since I've been involved in this so it's better to talk with HMRC.