I agree with richard_smith237. You kids should exit on their Thai Passports. Do not show immigration a TM Card or anything that would link them to the kids British Passports. Your wife and kids will not need to extend their Visas as they are Thai Citizens. As for you, you can extend your tourist visa at the immigration office. I think it is around 2,500 baht.
My kids came in on a USA Passport and got this Thai Passport a year later. Last time we left Thailand. We left on his Thai Passport.