I have an METV. When it is 4 months old (and hence still valid), I will be returning to the UK for a month. I will get a new passport in the UK. I will then come back to Thailand.

 

Therefore the METV visa sticker, which is still valid, will be in the old passport but the new 60-day stamp at Suwannaphum will be in the new passport. How do I deal with this at immigration upon arrival?

 

And then at the end of that 60-day period, I will pay for a 30-day extension at an immigration office. How is that dealt with in terms of the 2 passports?

Carry both until old visa expires.

No problem on entry. You will use your old passport to show your valid visa and the stamp into the country in the new one. You will continue to do that until the visa expires.

When you apply for the extension they will do it in your new passport.  They will also do a stamp in it with spaces on it for info about your old passport and the visa used for entry. You will need copies of your old passports photo page and your visa.