If you only work for her on a regular basis you cannot possibly be self employed.
You can. Really you can. There have been times in the last 21 years of self-employment where I haven't worked for anyone, let alone one person, and times when I've only carried out work for one person at a time.
Working only for one person does not, in isolation, mean that someone has to be an employee. It is one of many factors that may be considered by HMRC or later on a court if it is that someone's worker status needs to be reviewed.
Other factors that may be taken into consideration (and all of these I have read / heard said that "oooh but if if you do that you aren't self-employed!) include wearing a uniform provided by a client, having training offered by a client, and working the hours requested by a client. There are others.
Ultimately, the word which gets used the most when deciding a worker's status is "control" and how much "control" the client had over the work. If, say, in the case of the OP she wants to be a freelance nanny, then it's down to the client to state what her needs are, and up to the OP to tell her what she can offer her, how much she wants for doing it (be it per hour or for each day/week/month/ per activity), what hours she is willing to work, what day, and on it goes.
Admittedly, if the client in the case of the OP is wanting to treat the OP as an employee and then hand back the responsibility of being an employer to that of the OP, then that's not on and they're leaving themselves open to greater opportunity for the OP to bring a case against them at a later date, to argue they should have been employees.
As someone mentioned further up the thread, Uber was one such company that had a high-profile case against them by their workers. Another was Pimlico Plumbers. In both cases it was found in favour of the worker that the court felt they should have been employees, however, in both cases it took a huge amount of time to consider it all and to arrive at a conclusion. In the case of Uber, it was certainly reported that not all workers were happy about this either - it was a certain number of them only who brought the case. Many seemed very happy as they were, picking and choosing their work.
While nannying is one sector where I've heard there's been a spotlight shining on it (so to speak) in respect of people being self-employed where they may otherwise should have been employees, it doesn't automatically follow that this is the case.
As before, there are many, many factors that have to be considered, and the only way this is going to be a problem for the client of the OP is if the OP brings a case against her. The onus now is on the OP to take control of the situation and be that self-employed person which she claims to want to be, by setting out her terms and conditions to her client as a freelancer.