I watch 2 kids one day a week for a mom on our board and I always bring my own lunch. I don't think we ever talked about it. She always tells me eat whatever I want but I would feel funny eating their food.
I would assume I'd feed the nanny unless she had some dietary restrictions. Food was always provided for me when I babysat/nannied [of course, I was a lot younger then and probably wouldn't have thought to bring my own food].
When we had a nanny we provided snacks and would have done meals if she had to stay late through dinner time.