What are the embassy letter requirements for transferring visa stamps to a new UK passport in Thailand?

No letter needed.

There is a standard letter that you can download in English and Thai that explains that a letter from the embassy is not required.

c05cff/Information_Note_Visa_transfer_to_new_passport.pdf

Don't worry, there is no longer a letter requirement. I just renewed my passport a few weeks ago, Just go along to immigration and explain you have a new passport, have copies of your new passport info page (with the photo) and old passport info page, also entry stamp and extension stamps. Remember, that you will also have to update your tm-30 otherwise online 90 day extension will not work. I have a form letter in Eng/Thai from the UK Embassy explaining that they have agreed that no letter is needed anymore, but the group rules do not seem to allow me to attach it here. There is a pdf, available that states this in both English and Thai, and I am pretty sure I got it from a leading member of this group, but I am not allowed to attach it or show it to you due to group rules.

As you have a UK passport, I believe your embassy no longer issues that letter with a new passport. You don't need that letter. It might be a good idea to bring the passport issuance receipt if you have one. There was a report a while back someone asked for it instead of the letter. Usually, the immigration office knows which nationality can have that letter or not. So it wouldn't be an issue if you don't have one.

I don't know where you saw "letter from embassy (if needed)”, but I could say "if needed" is not so correct explanation.