Would you like someone with greasy hair looking after your children? Hygiene is a fairly key point with childcare.
I really wouldn't mind. One of the women who worked at my son's nursery had unattractively lank and greasy hair, wore scruffy tracksuit bottoms and t shirt etc. She was lovely, kind, gentle, had sound childcare practice. She didn't smell, so I presume she was fundamentally clean, and beyond that I genuinely don't see the issue. The fact that her hair was greasy didn't mean she was carrying diseases, or wouldn't ensure he washed his hands after using the toilet, or whatever, so I am really still lost as to why it might make her unsuitable for working in childcare. In terms of role modelling the only things that matter to me - kindness, patience, conflict resolution, being interested in the world, having fun - she was excellent.
Running fingers through hair then touching things - yes, sympathies. Although again, that's true of anyone who uses any kind of product on their hair. I can't help inwardly wincing when my son cuddles up under my chin if his scalp has recently been oiled, because I'm instinctively like 'nooo! Get your oil off my poor spot-prone face!' but I have to manage that reaction, because I love that he's cuddling me and his hair needs oiling. In a way, this isn't the same thing (his hair needs oil vs your nanny's needs washing, etc) - but it amounts to the same thing.
In any other employment context, I still don't get it. Employees or colleagues smelling, yes, intervene. Whether a nanny/au pair or someone in a huge office. It affects you because having to smell something unpleasant is not nice. Genuinely don't see how unwashed hair is comparable.
Fwiw OP I think Orwell is right that it's other people's attitudes on this thread that are worrying, rather than yours - I don't think it was you who said to tell her to wash her hair every day, which is quite frankly awful for almost everyone's hair. And I'm sorry that you think I was being unkind or unhelpful by checking whether she was able to lock doors and make sure nobody could come in while she was using the shower - I meant it as helpful. I remember, when I was much the same age as her, being uncomfortable using bathrooms at other people's houses if I couldn't lock the door. Not because I particularly thought they would come in on me, but I felt vulnerable around it, it was an issue I had, and thought was worth eliminating this possibility in your case.