I think there's two different concerns in this thread that are being confused.
One of the reasons that women might want to request a female nurse or doctor is that men are much more likely to assault a woman in particular. A male patient doesn't have the same concern. We are typically physically weaker, our word carries less weight, more vulnerable and many of the abusers in professional settings are sexually attracted to women.
So on the one hand, we have the idea that women need protection from men, because men are way more likely to assault women. I think this is right. But it doesn't apply in the same way to children, because people who abuse children are not more likely to abuse little girls as opposed to little boys. Child abusers often abuse little boys too. Ok, so child abusers are also overwhelmingly men and perhaps it is a fair concern, on the basis of those few, to ban all men from any profession with children. But framing this problem as a woman's right's issue is a red herring. Child abusers don't only abuse female children.
The other, in my opinion unrelated, problem is this "dignity" issue of women having the right to dignity over not having their private parts seen by a man. I don't really see why this applies to children either as surely the background reason we want this dignity is the sexual violence that men perpetuate against women. It's seen as undignified because there's an unwanted sexual element to a man looking at a naked woman. But unless these men are child abusers that's not the case here. And if they are, we don't want boys around them either.
So perhaps OP it boils down to not wanting men in personal care roles. I think there's an arguement to be made for this, but it won't happen at nursery. Really this is an arguement for a childminder.