Hi have been there. It's a quiet difficult process. You really need a lawyer. Everything documents ect are all obviously in thai so you won't understand what they are doing. Does the lady agree it is your child? If so I think you just go and register your the father at the local amphur office. But I would still use a lawyer to be sure.
Ouch!! Can't find any flights until 10th and that takes you around the world for over 1000 pounds! Surely your best forget your flight from bangkok and book new flight from Cambodia to your country