Hi, I work in compliance for HMRC.
If the Nanny is providing you with a personal service (ie, they don’t have the freedom to send you a replacement person to do the work), you have control and direct them, they have limited financial risk (you cannot use a contractual clause to force them to correct errors in the work in their own time) and they only provide the service to you then you have yourself an employee.
They may still be an employee even if they work for multiple families. You can have several jobs.
If you have an employee then you should be deducting income tax and NICs at source. They’re entitled to minimum wage, holiday pay, redundancy etc.
The onus was on you to check prior to engaging and paying them.
Even if the Nanny wants to be treated as self employed and paid gross then that is not in the gift of either of you to decide.
I would get some advice starting with the CEST tool.
https://www.gov.uk/guidance/check-employment-status-for-tax
HMRC can definitely look at this retrospectively, raise assessments and look at penalty behaviours.