As the mom of a tot with tree nut allergies, this was my biggest concern of the trip, and I'm here to say that I have never been anywhere that was more food allergy-friendly.
When making reservations at restaurants in the park (which you can and should do online months before your visit), you'll see there is an option for notifying the restaurant about your allergy. Upon arrival, the maître d' always confirmed the allergy with us, asked us to identify the child with the allergy, and twice had the chef come and confirm the lack of tree nuts in our meal choices.
Even more impressive? When all the kids wanted an ice cream break (and believe me, it wasn't unusual to have more than one of these each day — it was vacation after all!), I was pleasantly surprised to find that every push cart in the park was supplied with a binder that listed the treats' ingredients in large letters, highlighting those that contained gluten and various nuts.
