Talk

Yes, as long as that nanny does not bring their own child to work and as long as they are not pregnant as best to avoid close contact (though NICE says "there are no known adverse consequences for the fetus.").

As an adult they can take precautions to prevent spread of infection - such as wearing disposable gloves to change nappy, promoting frequent hand washing, not sharing towels or eating/drinking containers.

Nanny can keep a log of fluid intake - dehydration can occur. They will also keep a medication log.

NICE: Hand Foot and Mouth, Preventing Transmission

When you say childminder won't look after them obviously, has she said that? I cm and had a child with this recently and after researching I found it wasn't on the exclusion list unless children were actually unwell with it, which this child wasn't.
www.gov.uk/government/uploads/system/uploads/attachment_data/file/353953/Guidance_on_infection_control_in_schools_11_Sept.pdf Page 5. (Also bare in mind I am quite strict on following these guidelines and on poorly children attending settings, so if child is poorly, don't send them till they are better).
But if it's Childminders policy not to care for them, that's up to her. But given that it's not an excludable illness I can't see that it would be a problem to ask nanny.

I wouldn't because I caught it from a child and I was 10x worse than they had been

You say " our" childminder, why can't your dp/dh look after the dc?

Ugh, I caught that from DD and I ended up so poorly I managed to get Bell's palsy and had a fortnight off work. Nobody else got it though, I think it's fairly rare to get it that bad as an adult. I'd ask your CM first, but I reckon if not, Nanny would be ok too.

I'd expect my nanny to look after the children in these circumstances.