you can get an extension of 30 days , if you're already on overstay , then you need to leave , pay the fine ( 500bt a day) and then you can return . UK citizens get 60 days visa exemption, everytime they enter Thailand, so no need to buy a TR visa.
You go through immigration first, before luggage. But it's a long walk at Suvarnabhumi. Go to the toilet. Hang about so you pass immigration after midnight