I don't really understand what they are looking for - unless its to guard against fakes? A marriage certificate issued in Thailand is a legal document and is recognised by both the UK and Thai authorities - I can't see why any other nation should want further proof.
Are you sure that its the marriage certificate they want legalising? Could it be that what they actually want is it translating and the translation legalised? In either case, the usual route is through the MFA - outside Thailand I would guess that the Thai Embassy is the only place you might be able to get it done. I think you need to check with the UAE as to what it is they actually want. I'd be interested in hearing the result of your endeavours.
It could of course be that they want it certifying (notarising or whatever it is its called) and in that case, a solicitor or Notary should be able to do that for you but again, I'd check with them. Firstly, a UK lawyer or Notary is likely to want the certificate translating and secondly, it is always possible that the UAE may ask how a UK lawyer or Notary can legalise a Thai document.
I have a gut feeling though, that its not actually going to be all that difficult - I was taking with a guy who lives in the UAE about a similar matter and he said they are a bit like Thailand - they ask for lots of documents but rarely check the quality/authenticity of them.