And again, with greatest respect, I beg to differ. The first two sites that you link two are trying to sell payroll services (so of course they're not going to help you set up a self-employed structure!).
The HMRC site does give some indication of the various items considered when looking at employment v self-employment. I would totally agree if you had said that having a nanny working for your family solely for a 60 hour week in your home under your terms can be nothing other than a contract of employment.
However there are many other ways of structuring a contract for nanny services, especially if you are only looking for part-time and flexible cover. These won't suit everyone, and probably wouldn't suit the OP, but they can exist such that a nanny is not an employee and it is not illegal.
Yes, I am one mother who was happy to have a nanny who wouldn't (and didn't) agree with every request that I made in terms of hours, and who could send a (pre-approved, qualified and known to the children) alternate. And this arrangement worked very well for us for 2 years and was discussed with the tax office. The nannies (who worked together effectively) worked for a number of families (most of whom are friends of ours). There are of course increased risks to both sides - the parents may not get the nanny they want on the day they want, and likewise the nanny may not have work on every day. But of course this is one of the risks of self-employment! But given that most of the parents involved were self-employed and could themselves be flexible this arrangement worked very well. In fact it continues today though I don't tend to be part of the arrangement.
So once again, whilst the majority of nannies are employees, it is legal for a nanny to be self-employed. It is just rare that the circumstances allow her to be. And I would point out that the above arrangement came about as a significant number of us needed a nanny (and preferably the same one, or possibly one of two) for irregular numbers of days. We weren't trying to avoid tax or NI, and I'm not sure that this arrangement was particularly cheaper on an hourly basis say.