Without knowing how severe the hypothyroidism was, it is difficult to know -- if mild-moderate, then should be able to do most anything he wants. If severe, it may take a few weeks to begin to get levels normalized.
Both my boys have Hashimoto's. They are 12 & 14. My youngest has had hypothyroidism for two years. His antibodies are in normal range since he has been on medicine. He still plays football & baseball.
The doctor just started my oldest on medicine because his antibodies are so high and this will help bring them down. He isn't hypo yet, but the doctor said his TSH is going up, but still in the normal range. He plays baseball & football also. Started his med during football last month.
The doctor told me they need to keep an eye on how they feel. They haven't had any problems playing sports.
It has been a battle keeping my youngest son in range. Med keeps having to be increased since he is still a growing boy. He has good days and bad days. Not easy to deal with. We also just found out his ferritin, vit D, B12 & zinc are all low. Trying to increase with vitamins.
Good luck!