I had a nice reply and it got deleted when the power went out. Let's try again...
We went to WDW last May with TN/PN allergies (and vegan) and had no trouble finding safe food to eat. Almost every quick service location had binders with ingredients and knowledgeable and helpful staff. We were able to get veggie burgers and fries from Restaurantosaurus in Animal Kingdom (Dinoland USA). It took a little while to get our meal as they had to make everything on a separate grill and dedicated fryer, but it was worth it. They had special buns too which were okay. We visited Sunshine Seasons (The Land Pavillion) at Epcot A LOT, so much so that we even took the monorail there from Magic Kindgom. They made special plates in the back and had a wide variety of Enjoy Life chocolates and safe treats. The Main Street Bakery in Magic Kingdom (Main Street) as well as the food court at Pop Century had some cookies that were safe for PN/TN and a few other allergens. We had a poor experience at Pizza Planet in Hollywood Studios as they took a really long time to tell us what would be safe, said there were no nuts on the premises, and then later told us they had cupcakes with nuts on them. Not cool. Every other quick service place we went to was great though. Oh, and the burrito place outside Mexico at Epcot could make PN/TN-free burritos too. Sometimes they have nuts there, so it's not safe, but most of the time they are PN/TN-free! There was also vegetarian chilli somewhere, but I cannot remember where. I've also been told that turkey legs and pretzels are safe for most allergens, so it's worth checking out. We also were told that Dole Whip was safe but we couldn't find any the past two times we were there - so sad.
We had a very bad experience and way too close call on one of the tours that included lunch. I was terrified to see whole nuts in my meal that was specifically labelled PN/TN-free! I was traumatised for the rest of the day. Disney apologised profusely, refunded our tour ($200/person), and told us they would revamp their whole allergy policy for tour groups - and they did! My mother went back a few months later and when she booked her tour with allergies they told her they had new allergy procedures in place. It was like night and day compared to my trip. She had a wonderful time on the tour and had a great lunch free of her allergens. Their new procedures reflect the allergy attention we expect from Disney, so I'm happy that has improved dramatically.
Otherwise, we had incredible service at all the sit down restaurants we went to, as well as the Pop Century food court. It was heartwarming to hear chefs tell me 'it's my pleasure to make something special for you', 'that's an easy accommodation' and 'I love taking special requests'. That's some real Disney magic!
You should be fine. There are a lot of safe foods and treats for PN/TN allergies. Have a great trip!