As stated above, you probably need the marriage cerificate legalized in the uk.
It was very hard to do ten years ago when i considered it first, but a bit easier now that i needed it..
I have just done it.
You can now order legalization by the Uk FCDO office and that saves a lot of the earlier problems.
Go to https://www.gov.uk/get-document-legalised
it explains what you need to do.
You can now pay online and order that docs are sent to an address.
You must send original copies of cert to FCDO in UK.
Obviously, be careful how you send it if its the original.
I had a few original copies from the uk registrar, so i did not have to send the main original.
Once UK legalized, it needs to go to the Thai Embassy in London for Thai legalisation.
I was visiting the uk when I did this, so I got the Uk legalisation sent to a friend near london, picked it up and went to THai embassy walk in morning ( no appointment - took 15 minutes). They said it takes 3 days. I paid and gave them a stamped adrressed envelope registered first class and received it in Manchester 4 days later.
If you are not in Uk then you need somebody there to do processing.
attached FYI, Copy of legalisation on back of marriage cert.