Not easy for Cambodians to stay long term, unless they have an employer who can use the migrant worker scheme, but they need to return home when they're work is done.
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.