As Tod Daniels says, UK passport renewals are done via VFS in Bangkok and actually issued by HMPO in Liverpool. It has nothing to do with the British Embassy.
I managed to get Re-entry permits at Chaengwattana for my wife and son without them being there, but I had to be referred to a senior officer (and grovel) to do so. The bonus was they were issued straight away so I didn't have to wait in the queue to collect the passports.
You just need a copy of your passport. But you need your long-form birth certificate, proof of address in Thailand plus the mother's ID/passport/birth certificate.