I was thinking more in terms of the equipment being used for nuts, and that sometimes actual nut product can make it into a non-nut product, if processed in the same facility.
Well, at least the allergic child isn't in your daughter's class! But, it does seem quite overboard to ban nut products in the entire school. However, on the other hand, I guess it could be a concern if someone added peanut butter to an item being served at lunch that would not normally be suspicious for having nuts.
It is crazy how things are today. Years ago, if a child went to school with an allergy like that, the child would merely bring her own food and be taught not to eat food that other people bring. But now the entire school has to change policy! Hope that does not create a false sense of security for the child. Mistakes can still happen.