You can get to one of the stickies mentioned by the other poster from a link in my signature to the disABILITIES FAQs thread.
Post 3 of that thread has information about different conditions, including food allergies. Nut allergies are actually one of the common food related issues they deal with, so they have a lot of experience.
At counter service restaurants, food allergies are handled by speaking to a lead CM (Cast Member = Disney Staff). They will have an ingredient book or list of items in their menu that include that allergen.
At full service restaurants, the chef will come out during the ordering process to discuss safe menu choices. At buffets, the chef will go thru the buffet and point out the safe and unsafe items for the allergies. If someone feels unsafe eating foods from the buffet because of cross contamination, they can also bring out a portion that has been plated in the kitchen.
We traveled recently with a child who had multiple anaphylactic allergies, including peanuts and tree nuts, eggs, milk, shrimp and seafood. The process worked well and, despite his multiple allergies, we had very little extra wait.
We are at multiple buffets while there and the parents felt safe with the foods the chef pointed out, so they ate off the buffet.
There is also a thread near the top of this board about the
DAS (Disability Access Service) which includes a DAS FAQs in the first post. I mention this because many people coming to this board may have read on the internet about the old program (Guest Assistance Card = GAC) and are expecting that. GAC no longer exists - it was replaced either DAS.