After birth it takes approximately 1-2 days to register the baby according Mexican law . Your child will be immunised during the week after birth. Mexican Airlines do not allow flying with a baby who is below 8 days old.
The intended parents names can be included on the birth certificate. The surrogate must be Mexican. As a Mexican citizen such children area also entitled to a Mexican passport. However this will be difficult to obtain for same-sex parents.
Depending on your origin you may need to fly to your embassy with the baby for DNA testing. Some nations don´t have to do DNA testing