I had two vaccinations in England and one in Scotland and have a similar issue with the English NHS pass recording my booster (from Scotland) as a first dose so I have two first doses and a second.
The English covid pass people are IMO totally useless as the people on the phone can’t help, they can’t/won’t send you a letter confirming vaccination status and when you log a call with the Vaccine Data Resolution Service (VDRS) they NEVER call you back. I’m waiting for a single call despite raising two issues with them, the first over 2 months ago.
The solution for me (probably not suitable for your son) was to ask the fantastic, helpful and responsive Scottish COVID vaccine people to add my English vaccines to their system which they managed to do over the Xmas and New Year period. I therefore have a fully functioning Scottish pass with all 3 vaccines correctly recorded which works for the very strict Austrian requirements (so far anyhow).
Maybe worth asking the Welsh people to add the English vaccines to theirs?
On that note, I think it is absolutely farsical than one country has at least vaccination certification systems, totally unjustifiable IMO