How long will it take for my 8 year olds stomach to feel better after eliminating dairy?

Soon. If he is lactose intolerant, it should improve quickly. Read labels, there is lactose in alot of foods.
Depends on transit. The duration of symptoms triggered by lactose on the intolerant gut is time targeted & related to the time it takes for a given lactose exposure to transit the length of the intestine. In most this will be < 4 days (2 to transit the small bowel). So 4 days after starting the lactose free experience you should see some improvement (if that is the issue).If no better after a week, re-evaluate.