WDW is actually very accomodating and allergy-friendly in general.
The common cookies and other items are probably not to be considered safe, just because they may be prepared or stored next to nut items. However, there are readily-available nut-free packaged baked goods all over the place. A friend's son had several of the nut-free brownies from the Main Street Bakery. There are prepackaged (sealed) nut-free and gluten-free items all over the place.
There is a new Babycakes location in Downtown Disney, but they will no longer claim to be nut free since coconuts were classified as tree nuts, and they use coconut and coconut oil.
We traveled with a GFDF kid several years ago, and he was fine - they will work with that one pretty easily, he got GFDF pizza and several other things in various places. Dessert was, naturally, a little harder for him but he did ok.
You can email
specialdiets@disneyworld.com, and they will send you ingredient information, restaurant guides, and all kinds of menu information. If you are eating TS, you can have your allergy noted on your reservation and one of the chefs will walk you through the menus to make sure you are safe. Most TS can provide tapioca bread/buns and rice pasta, if necessary, and the chefs can also give you a safe/not-safe tour of the buffet if it's a buffet location.
