Two of my kids have peanut, tree nut, and shellfish allergies. We've never had a reaciton at WDW. The chefs are amazing at handling food allergies. We ate at Teppan Edo a couple weeks ago. The manager talked to us about the kids allergies, then when the chef came, he double checked. He cooked their food first. The manager even came back after the food was served to make sure they were still o.k. At buffets, the chef will walk you and your child through and let you know what's safe and what's not. My kids usually get special off-buffet desserts, because the desserts almost always have a problem.
Contact the general special diets email about 2 weeks prior to your trip. They'll email you a form that you fill out with your personal info, resort, ADRs and allergies. They take care for forwarding everything to the chefs at each restaurant. When you dine, the check-in CM will usually confirm the allergy, but mention it if they don't. Also, confirm it with your server. Sometimes the seating CM forgets the allergy note. The chef will meet with you at your table and discuss safe options and any modifications that might be necessary before you place your order with the server.
Have a wonderful trip!