My DD also has egg, peanut, and tree nut allergies. We enjoyed Divvies throughout our vacation. We got them at almost every restaurant. If DD didn't eat them, I kept them in my purse for easy use later on.
I can't remember about the chicken nuggets. You will need to talk to the lead at the restaurants to find out about ingredients.
At most breakfast places, we ended up having their standard gluten free, egg free, peanut free, etc. free waffles. They didn't have an simple egg free alternative. Instead, they seemed to have one meal that serves all major allergies. I didn't like their taste, but DD didn't mind.
Be aware that we waited a long time (longer other patrons) at table service meals because the chefs had to come talk to us, then make DD's meal from start to finish. It was worth it for her to be safe, but for a 20 month old, that might be tough.
I would call sooner rather than later to be sure you talk to the necessary people.
Have a great trip!