Plenty of solid advice on here about why it's vanishingly unlikely nanny would meet HMRC's definition of self-employed.
Assuming you're in the UK @Countessofgranthamm and an assessment by HMRC would prove your nanny of 3.5 years should actually have been on the books as an employee, fag packet estimate of what you owe her as a thank you on termination of her employment because you're moving abroad is:
based on an assumption of £30k/year paid cash in hand and the nanny has actually paid all her own employee NI & income tax due AND HMRC would allow that to be transferred across to cover those obligations when correctly assessed as an employee, you would still owe something around the order of
£10k in employers NI for 3.5 years
£2.5k employers minimum statutory pension contributions for 3.5 years
£2k in redundancy pay - exact amount dependent on nanny's age
potentially also £12k in missing holiday pay and
£5k on grossed up er and ee NI, income tax and pension contributions on the unpaid holiday pay
Sorry that's not the answer of 'yeeeeah, a 200 quid john lewis voucher & a bottle of pop will be fine' - which is presumably what you were after OP?
Hope your earnings in whichever tax haven you're buggering off to work in will cover what you owe in back-pay to the UK coffers and your nanny!