Having done some googling, I believe your son’s watch has a Calibre S movement, which appears to be an in-house design, remarkably.
Cousins are unhelpful, but it may be worth calling Gleave and Co in London (0207 253 1345).
The only stem I could find on eBay is in the US - Stem for Tag Heuer Calibre S https://rover.ebay.com/rover/0/0/0?m...2F254173818249
It’s not cheap, but it does have the crown which you may need, depending on where the stem has snapped on your example.