What happens to my Thai 'non-O' visa and re-entry stamps if my passport expires while I'm back in the UK?

Embassy gives an official letter saying about the passport change; Thais officer are used to it .

Think you carry both to show immigration. Not sure if passport office gives you anything else to show. Think you then transfer your extension to new passport. Wait for admin.

First of all, there is no such thing as a 12 month non-O visa. All non-O vias are 90 day visas that grant a 90 day entry to Thailand. Once you enter, you can apply for a 1-year extension (not a visa). An extension will never go beyond the validity of your passport, so if you apply for the 1-year extension and only have 6 months left on your passport, you will only get a 6 month extension. So the premise of your question is about a situation that cannot occur.

If you renew your passport while you are outside of Thailand while your extension is still valid with a re-entry permit (renew early, run out of pages, etc), then you would need to return to Thailand before your extension expires in order to be able to re-enter using that extension. You would use both passports when you re-entered, and then go to your local immigration office to have them move your stamps/extension to the new passport.