They will need your Long Form birth certificate to prove that you are British Otherwise than by Descent. Also their Thai birth certificates, translated into English. Everything else should be straightforward.
Although your son is on overstay, children are not fined or banned, so there isn't an issue with him remaining in Thailand. When he leaves Thailand, you should try to get him proper visa to return. Unless he's likely to remain in Thailand for a long time and go to school here, it's not really an issue if he doesn't have a proper visa.
Children who overstay are stamped as such in their passport. They are just not fined or banned. Having an overstay may cause issues if they want to attend school in Thailand.