My fiancee has a pretty bad peanut and treenut allergy. In the 3 or so trips we have taken together, there has never been an issue. I note it on all of our ADR's. In just about all cases a chef has come out and given her a talk about what she can or can't eat or any other options they may have that are not on the menu. In cases of buffets the chef has taken her (I like to tag along) around the buffet and goes over every bread, dish, sauce, dessert, etc. and what is safe for her to eat and what she should stay away from. Usually the chef will come back towards the end of our meal to see how she liked everything or will go over the desserts with her if they didn't already.
We don't usually snack much. We usually stick with ice cream, ices, soft pretzels and popcorn for the most part and she has never had an issue with any of that.
Just make a note of the allergy when you make your ADR's and mention it when you check-in (if they don't mention it first which they usually do). Disney is really impressive when it comes to catering to those with allergies.