@AnonSW20 I think you've misread a quote here - it's didimum who had the self-employed nanny. I was the one who did a rough calculation of what NI, pension, holiday & redundancy back pay may be due.
In didimum's circumstances, it does actually sound like they'd been VERY careful to ensure a very part-time nanny could genuinely be considered self-employed. (Rightly so if one of them was working for HMRC!) There are not many families out there who can have a nanny for, say 9 hours a week and give them the flex to say 'I'll be doing 3-6 M,W,F in week one, 3-7.30 Tues & Thurs in week 2, and 9 hours on a Saturday in week 3 on a rolling basis - which is essentially the kind of latitude needed for a nanny role to be considered genuinely self-employed.
didimum had made a great song & dance about no-one knowing what OPs arrangements are, as they "might" be this flexible. Although plenty of early posts did ask whether the nanny could set their own hours etc, but OP chose not to answer.
Meanwhile the reality is that the overwhelming majority of families using a nanny will have fixed days & times they need during term (even if only 2 regular after school days a week). Or if not, will still want the ability to set their hours around their own shift patterns. Hence aaaaaall the comments from those working in tax that nanny should almost certainly be considered an employee & taxed, pensioned & redundancy paid as such.
So I think we violently agree @AnonSW20 ;)