You just apply for her UK passport via VFS in Sukhumvit soi 13. You will need a translation of her Thai birth certificate (showing your name) but it does not have to be stamped by the Thai MFA. Most importantly you need YOUR long-form birth certificate showing that you are British Otherwise Than By Descent. You will also need the mother's passport/birth certificate/ID and, if you're married, your wedding certificate (translated if in Thai).
If you're getting married in the UK, get the marriage certificate legalised by the UK Legalisation Office in Milton Keynes, then certified by the Thai Embassy in London, before you come to Thailand.