R.,
I think it depends entirely on how sensitive your child is and what the potential consequences of exposure are. You really should consult an allergist because allergy testing of infants is notoriously unreliable.
My son is now 2 and he's highly allergic to both dairy and soy proteins. As an infant, he had eczema, bad reflux and didn't sleep until I eliminated all dairy and soy from my diet (both obvious and hidden in other foods - I was nursing). Now that he's older, he still doesn't tolerate either food and has issues on and off with the eczema. Soy will give him diarrhea and hives. Milk gives him diarrhea (gray area between intolerance and allergy). When we had him tested for food allergies all of the tests were negative - they said he's not allergic to anything. But I can tell you from hard experience both dairy and soy make him pretty sick. We are still avoiding all obvious and hidden dairy and soy in his diet. He's sensitive enough that he can't tolerate even small amounts of dairy or soy cooked into other foods. BUT, we've never had an issue with cross-contamination. He eats things processed in factories where dairy and soy are used and there doesn't seem to be any issue.
That said, nut allergies can be deadly. I'd definitely talk to the doctor. Although diarrhea and hives are unpleasant, I'm not concerned about anaphalactic shock and my son dying if he gets exposed to dairy or soy. If my son had a nut allergy, I'd go out of my way to make sure he wasn't getting even trace amounts of nuts in anything he eats.
I don't know what kind of products you are using on your son but I've been using California Baby super-sensitive baby wash and it works really well. I use either Eucerin cream or Aquaphor on him. I've found that in addition to the food issues, the products I use on him make a big difference to his eczema. Some other things that have really helped him: getting a HEPA air filter for his room and running it all the time (I got it at home depot for about $130), washing his bedding once or twice a week (I replaced my washer with an HE -high efficiency- maching and that has REALLY helped), keeping our pets out of his bedroom and minimizing the use or strong chemical cleaners in our house. I've found that my sensitive child is sensitive to many things.
Just in general, I've found the easiest way to deal with a dairy and soy allergic child is to go to a primarily "whole" food diet. We eat lots of meat, veggies, fruit, potatoes and rice. The more processed the food, the more likely there is something in it that will make him sick. Since convenience foods are pretty much out I tend to cook bigger dinners and then we eat it twice in a row or I'll freeze half and use it later (I'll do chicken kabobs one night and then serve the same thing the following night with tortillas, refried beans and call it fajitas!). He also tends to get whatever we have tonight for lunch tomorrow. In my experience, reading labels will make you crazy because they reformulate stuff all the time. What is okay this time may not be okay next time.
:-)T.