My daughter has a peanut allergy and believe me, Disney knows their way around this and all other allergies!!!! I just called for my ADR's and told them we have a person with a peanut allergy and on that trip we also had a strict vegetarian and a low-carber and they made wonderful meals around it...
We just told them about the allergy and never worried about it again and on each of our reservations they had it noted and asked my daughter not to order or go up to the buffet until the chef came out to talk to her. The chefs at each restaurant would come out and talk to her about each and every menu item and each dish on the buffet. Many times, they either suggested a certain dessert or offered one they "specially" prepared for her. Some desserts are bought and brought into the restaurants so they would say because there are other nuts in the dessert, they would recommend her not to order it because it MAY also contain peanuts. I don't think you'll have much of a problem at all if you're certain to let them know with your reservations..... AND each time I called to change a reservation or add a reservation, I made sure it was still noted.... Then there was nothing to worry about after that.