Just returned from a week at WDW with a peanut allergic child, age 5. I have to say that Disney really handles food allergies well, and everyone we encountered was knowledgeable, kind, and attentive.
I made note of dd's peanut allergy on all of our reservations. I also contacted Brenda Bennet at
Brenda.Bennett@disney.com from their food services via email and heard back from her within the day. She sent me a list of all snacks sold on the premises and what potential allergens they contained. I was also given a form to fill out which was sent to all of our TS restaurants.
We mostly dined at character meals and we very much appreciated the buffets. We met with the chef at every meal - s/he came out to meet with us very quickly on all occasions to go over the menu. If it was a buffet, they would walk the buffet with me and point out anything that might not be safe. We were so excited that the ice cream bars were soft serve, and even the jimmies and sprinkles were safe! On one occasion (Askerhaus Princess Breakfast) the chef even offered to make a pancake for DD that was not on the menu.
Our food court was also very accomodating - we stayed at the Port Orleans French Quarter. We were told that there were no peanuts in anything at all (maybe no nuts at all? I'm not sure as we do not have a tree nut issue...) The only thing to stay away from was the pastries. And the homemade ice cream was SAFE!
We also have a dd who cannot tolerate cow's milk. We were pleasantly surprised to see that the food court kitchen and almost all table service restaurants were able to serve her soy milk.
BTW, I've been told the Mickey ice cream bars are a "may contain," so I'd be extra careful with them. We stuck to the Itzakadoodle popsicles to be safe. Oh, and some places sell packaged cookies made by Divvies; look for them - they are nut free!
http://www.divvies.com/facts.php
Good luck! I have to say that Disney is one of the few places I feel comfortable dealing with dd's allergy.