I think it’s all been covered above.
The Thai authorities advise you to get a yellow ‘vaccination passport’ for international travel. In Phuket it comes from the Gov’t health office in Phuket Town.
However, the UK Passenger Locator Form just requires one to declare that one is fully vaccinated under a recognized scheme (which Thailand now is).
My experience of a recent flight to London is that no one in UK is interested in looking at your proof of vaccination (at least certainly not pre-Omicron).
If you can get KLM to accept the Thai vaccination certificate provided by the hospital (best done at check-in), it’s probably job done.
Ironically, in a reversal of this, my impression when I applied for the Thailand Pass online was that the Thai authorities might not have accepted the (Thai-issued) yellow vaccination passport ????. Luckily I had my Thai hospital vaccination cert with me