I don't see anything wrong with describing the colour of someone's skin, though I think you're probably right in guessing that nursery might disagree. There's lots of stuff online if you google for how to talk to children about race. I think it's really important for parents of white children to do this (parents of non-white children can hardly avoid it).

I hate hate hate that "you mustn't call people fat" thing. It just reinforces the idea that there's something wrong with being fat.

I'm not sure what the answer is to the bigger question of nursery telling her one thing and you telling her another. From things you've said previously, it doesn't sound like they're enormously good at dealing with differences of opinion. I do wonder how they deal with, for example, questions about religion — whether they're willing to take the line of "some people believe X, other people believe Y". When (not "if") Tobes comes to me in a few years time and says he's been told it's bad to be fat, that's the line I intend to take.