When you email they will email back a list of safe snacks. I was quite surprised at the number of peanut/tree nut safe snacks, there were a lot more than I was expecting. There is a chocolate brownie that is safe and it was absolutely delicious.
I did find it a little harder at the restaurants when it came to desserts. All of the restaurants were wonderful and there was always something available or something that they would make for me that was safe, but I really didn't end up eating a lot in the dessert department. Dessert to me is chocolate, sweet, gooey, yummy, naughty and fruit doesn't cut it for me

I like fruit, but its a snack or a breakfast item, not a dessert. From what I remember all of the yummy desserts that I wanted off the menu were not safe, that doesn't mean there weren't other desserts that were ok, just not the ones I wanted

We did eat at mainly TS restaurants on our last trip and we were there for 21 nights and I honestly don't remember any desserts, so if I had one it was not that exciting. The brownie on the other hand was delicious and that became my daily sweet treat.
I have travelled and do travel a lot all around the world and WDW is the BEST place I have ever been when it comes to allergies. It is somewhere I really don't worry beforehand about where I will eat or doing research because I know when I am there, I just mention my allergy and it will be looked after and taken seriously (well I do look at restaurant menus and decide wether I will eat there or not).
Enjoy your trip
