It's a lot cheaper and potentially quicker to get a new British passport in the UK than it is from Thailand, so I wondering whether it was possible to get one's extension of stay and re-entry permit transferred to a new passport at a Thai embassy or Consulate outside Thailand?

Or is that even necessary? Can one re-enter Thailand showing both old and new passports and using the re-entry permit in the old passport, then get the extension of stay transferred at one's local Immigration office?

No only immigration inside Thailand can transfer extensions of stay.

Yes you can enter using re-entry permit in old passport and be stamped in on current passport.

It's a lot cheaper and potentially quicker to get a new British passport in the UK than it is from Thailand

Its only cheaper if you happen to be in the UK for another reason...not cheaper to fly there just for the sake of renewing a PP...