Nanny here. I wouldn't mind my nanny family to fly business and have me fly coach. Flying business requires many $$$ and I'd feel awkward to have a family pay that amount for me to fly business unless they were super rich of course.
But I'd be upset of taking care of the child on board as coach seats are so tight, it can be a nightmare, especially if the kid knows their parents is onboard as well - and they will absolutely know they are, we're not playing hide and seek
Anyways, I'd just be happy to go on vacation, wherever that is and find out about a new location. I don't care about not being offered business class.