Two kids with nut allergies here. One is allergic to tree nuts (all nuts/seeds other than peanut). The other is allergic to peanuts. The peanut allergy is severe and we carry an epi pen. They are both also allergic to other foods, including dairy, egg, soy, canned meat (no idea what this actually is but he can't eat canned tuna or chicken or anything).
For the egg white allergy: I would avoid giving him egg yolks too. You can not get the white entirely off the yolk. My oldest used to be just egg white allergy but now is sensitive to both parts and I think it's at least partly due to the exposure to bits of white on the yolk when I did give it to him. Also, be aware that vaccines can be made in egg or with egg - specifically flu and MMR. They will tell you to give Benadryl beforehand or have you stay in the office after to watch for a reaction. We ended up stopping these vaccines since I noticed an increased reaction after the first one. Didn't want to expose them further.
Peanuts: there are many related allergies. Make sure your son isn't allergic to those too. Ours has problems with garbanzos, peas and lentils also but not the other related foods. Google peanut allergy and the related allergens should come up.
Our kids can eat food made on the same equipment as their allergens so double check that like lazy-saver said.
Talk about it to everyone, esp your child and everyone that takes care of him. Don't ever assume that they know or remember. And with your child, he will pick up on the fact that some foods aren't safe faster if you talk about it every time you grocery shop, eat, prep food, etc.
Substitutions:
Many nut butters available.
You can use 2T of water per egg in most baking, like muffins, coffee cakes, etc. When the water doesn't work (like in cookies), we use Ener-G Egg Replacer. Works great but it's expensive so we use the water whenever possible. You can use flax or chia too but, again, it's more expensive. My kids can't do any eggs at all so...
Make sure you read food labels EVERY TIME you buy the food. Manufacturers frequently change their formulas.