The essential answerThe Thai part is quick. The destination's rules are not.
Thailand's own export process happens in the last few days: you apply on Form Ror.1/1, take your pet to the Animal Quarantine Station (AQS) for a health check no more than three days before departure, and receive an export permit (Ror.9) and a health certificate for the country you are going to.
What takes time is everything the destination country demands first. Many countries treat Thailand as a rabies-risk country and require a rabies blood test and a waiting period of three months or more. Some, like Australia, do not accept pets directly from Thailand at all. So the first step is to read the destination's rules — ideally many months before you plan to leave.