This is something you should have discussed with your nanny when you hired her
You're in a tough spot, OP. I don't expect my boss to pay me for a holiday day if she doesn't get the day off, but I do ask to be paid if she's off (and therefore I'm off) when holidays fall on my regular working days. I think most nannies over 20+ hours a week expect this. BUT, there are always arrangements to be made and compromises that can be found, it's just a matter of going into the working partnership with everyone on the same page.