bringinghomethebacon - There has been a huge amount of discussion about that particular topic over the past couple of years on here. Someone may be able to provide a link to one of the main message threads on this topic.
Conclusion to which was that in 99% of cases a nanny is an employee.
Employment status is looked at in 'the big picture', so taking into account all of the work done by the person say over a typical month. It also looks at what each job involves - what the relationship is between the person doing the work and the person paying for the work. A nanny working for one family would be seen as their employee, particularly if they are being told what time to arrive, leave, what to do, when to do it.
If the nanny worked for say 6 different families a month and was free to choose when to do that work, then they may be seen to be providing a service. Even then they may be seen as having an Employer Employee relation to each individual job, so a Status Officer may decide that each job is actually of Employed Status.
Employment law exists to help protect employees. It gives employment rights, including things such as Holidays/Time Off, Statutory Maternity Pay, Statutory Sick Pay.
A cleaner or gardener may have many clients. They may see several clients in a single day. Whereas a nanny usually works for just one family.
Not sure I've explained it well enough... perhaps someone else can phrase things clearer.