I think you should say that you need time off for a personal reason, and leave it at that, for two reasons. First, naturally thin people are , in my experience, REALLY not supportive because deep down they think that it should be easy to just put down the fork, and we know it is not that easy. Second, as an employer of a nanny, because it is such a close and personal job, sometimes both the nanny and the family really don't need to know every little detail about eachother's lives.
It's okay to just make up your mind to do it and avoid those who might not be supportive.
They would probably be happier if you found someone who could cover for you for the days off.
I did it on Monday and came back to work Wednesday, and didn't tell anyone at work. It all went fine, but my job is less physical than a nanny job.
Hope that is helpful.