I agree about being extra careful with WS restaurants, and the language barrier. That is the only time we've had a bit of concern for my son (he's dairy and egg allergic). He loved the QS place in Japan, and their chicken and rice is always safe for him.
For CS places, we have had good experiences, sometimes over the top! At AK, we ate at restaurantasaurus, and he couldn't eat the kids' chicken nuggets (this was years ago). The manager came out and swapped a hot dog with no bun (he couldn't eat the buns) and she went in the back to make sure they also changed their gloves when handling his food (he was also touch hypersensitive). The next day, we were there again at lunch, and the same manager was there, AND she not only remembered us but also what he wanted to order!
He's never had one reaction to food in all his trips to Disney!
For nuts, I am sure their policy is new gloves, but you can ask.
Desserts will get tricky, as once on the dining plan (before the new change) we'd just get the packaged cookie (he couldn't eat it) and bring it with us. Some ended up tossed (what a waste).
I think there is a kiosk at AK (or there used to be) that sold safe snacks for many common allergies!
At TS a chef always came to talk to him, and after explaining what on the menu was safe, he or she would ask what my son wanted them to make for him (off the menu a bit).
You should be safe, but bring liquid benedryl and your epi-pen (duh!).
By the way, Walgreens now carries the individual teaspoons of liquid benedryl (their brand) now. My son carries a few of them with him always, in case of slight reactions.