There is no specific case law (at least not any I can find) regarding nannies and babysitters in terms of employment status.
The main issue is one of Control. ESM0516
With a Childminder it is clear that a parent does not have control, as the childminder provides a service from their home, under their terms.
When care is at the parents home, control is an issue - as it is not about if control is exercised but it there is a right to exercise it.
In Business On Own Account - is another factor considered. ESM0514
A babysitter could possibly be considered to be in business on their own account but other factors would need to be considered as they don't have stock, premises, staff etc.
Personal Service - ESM0530
Again a nanny or babysitter would usually need to provide the service personally, they are not likely to be able to employ someone else to do it for them.
Mutuality of Obligation - ESM0543 Cornwall County Council v Prater [2006] EWCA Civ 102 In his summing up Mummery LJ said:
“Nor does it make any difference to the legal position that, after the end of each engagement, the Council was under no obligation to offer her another teaching engagement or that she was under no obligation to accept one. The important point is that, once a contract was entered into and while that contract continued, she was under an obligation to teach the pupil and the Council was under an obligation to pay her for teaching the pupil made available to her by the Council under the contract. That was all that was legally necessary to support the finding that each individual teaching engagement was a contract of service.”
Reading that you could conclude that a babysitter or nanny would be under a contract of service, as even when there is no obligation for you to provide work, or for them to accept an offer of work, when work is actually done there is a contract entered into where the babysitter is obligated to care for your child/children and you are obligated to pay them.
Length of Engagement - ESM0548
Not a big relevance. "Where work is offered and accepted occasionally and irregularly there is unlikely to be a continuous (or ‘umbrella’) contract of employment. Instead, each engagement may itself represent a contract of services or a contract for services."
Employment status therefore is very hard to determine when someone works very irregularly for numerous clients. The Big Picture (ESM0556) has to be looked at. Without the existence of Case Law it is hard to know if a babysitter is genuinely self employed, it could vary between babysitters as some may work for the same clients again and again, similar days and hours each time. Others may see a client once every few months, may go for long periods without seeing a client.
It is a grey area... no definitive answer in the case of a babysitter.
In the case of a nanny, it is a bit clearer as they are providing work to the same client repeatedly and the client is able to exercise control, regardless of if they decide to do so or not. The nanny is working for one, possibly two clients... they are not working for numerous clients. So the big picture points more to employment, than self employment.