Hi there Disfreek. We took our family to WDW in March and felt sooo well taken care of regarding dd4's peanut and treenut allergy.
Mind you, my daughter is very allergic, but not so much that she has had difficulty breathing when others have peanuts on airplanes. So, I did not worry too much about it there, just kept it out of our row. I also have a little bag that I have sewn with all her medication in it (albuterol, epipen, travel nebulizer, etc...) I also gave her some benadryl before we boarded, just as a precaution.
I was a little nervous about WDW, but they were amazing about it, very accomidating. The chef would personally come out and chat with us at each restaurant, taking us around and explaining what she could and couldn't eat. It was such a relief. They took it VERY seriously.
Hope this helps you out. Good luck!