I had the same issue with BB, after being assured my SIM and account matched. Turns out it was because my phone company still had my old passport number while the bank had my "new" one, issued in 2018. Easy to sort out after that.
It took about 2 weeks. However I had a family member in the UK send it to the Legalisation Office for me, and then the Embassy sent the final legalised document back to my family member.
Unfortunately not. The Thai Embassy in London needs to certify the UK Legalisation Office's stamp. Only the UK can certify that it's a legal UK document.