I've posted many times about the many positive dining experiences we've had at WDW with my two DD's who are allergic to peanuts and tree nuts. As others have said, Brenda Bennett is a lifesaver. She has even sent me ingredient lists for foods served in the other parks.
We've had some great and worry-free meals at several table service restaurants at WDW like:
Artist Point -- substituted the Berry Cobbler (an adult dessert that happens to be peanut and nut free) for the kids dessert
California Grill -- where they made my oldest DD a special peanut/nut-free dessert
Narcoosees -- super cautious chef came out twice to tell us not to let them eat the bread as it was made off-site.
Flying Fish -- very specific about safe and unsafe desserts. Also served my DDs Mickey Pasta, which was not an item listed on the menu.
Hoop De Doo Review -- kids just loved the show and the food.
Cinderella's Royal Table -- the server assured us that our children's breakfasts we're being cooked in separate pans.
Finally, if you ever visit other Disney locations, I think you'll find the same level of concern. At Vero Beach the chef at Shutters changed the children's meal dessert for the entirety of our stay there to peanut and nut free things. These desserts were served to every child, not just ours. However, my children's portions where stored in covered containers in a different part of the kitchen, out of the bakery area, to prevent any nut fragments from other desserts from dropping on them. We've also taken an
Disney Cruise, which although was more limiting food-wise for them (boy, did they eat a lot of fruit plates) was certainly safe.
I really think that Disney is quite serious about accommodating food allergies. I have found them to be responsive and cautious. That's why we keep going back!
DVCDawn