We are really, really fortunate that our nanny has never been sick. But she had a death in the family recently and needed time off. We took time off.
For snow days, DH is a Fed and they tend to either shut down or offer more flexible options for telework or taking leave. We've been lucky there as well as I think there's only ever been 1 time our nanny couldn't make it.
I don't think there's any option for back-up care that would be worth it for us, for the very rare instances we would have needed it. We also have no family in the area and I work in health care so technically I never get a snow day and can't take off when my kids are sick. I've told DH it will have to be on him.