My spouse and I fly all the time-usually United. In fact, I just flew three flights with them in the last two weeks. I am allergic to all nuts except almonds. United does not serve peanuts anywhere on the plane. That includes first class. When my husband and I fly separately, he flies first or business (both internationally and domestic) and has never seen a a peanut (he has flown almost a million miles on United and I have flown around 200,000 miles with them). In first, it is a tree nut mix containing almonds, cashews, and another tree nut. In coach (domestic) they sell a couple snacks that contain tree nuts. Internationally, they don't hand out bags of nuts. We have also had good experiences with British Air, Lufthansa, American, Virgin Atlantic, and Jetblue. My brother and his children are allergic to peanuts and have flown Hawaiian Air many times without a problem.
Have a great trip!