Hello... Yes, people, including children, can and do die from allergic reactions. Food allergy reactions seem to be especially potent and nut allergies are some of the most severe food allergies. And it doesn't even require the child to eat peanuts for a reaction to occur. Just being in a room with someone else eating peanuts or a peanut butter sandwich, for example, can trigger a reaction.
Not all allergic reactions are deadly, however. The kind that is that severe is called anaphylaxis, or allergic shock. You can read about anaphylaxis here.