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.
If the baby was born in Thailand there is no need to do anything. Obviously if the baby leaves Thailand at some point he/she will need a passport and you just have to present the Thai birth certificate at Immigration at the airport. We did this for our son, although on returning to Thailand we got him an O visa and now he's attached to my Extension. Even though there are no bans or fines for overstaying under-15s they do get an overstay stamp in their passport.
It should be no problem getting a 5-year or 10-year Visitor visa, if she has a record of visitor visas, with no overstays. All Visitor visas are ME, although she will only be allowed to stay in the UK for a maximum 6 months in one year.