Usually when they say that they waited several months, they mean that they had to wait to get an appointment with a specialist who either writes the T approval letter (a psych, therapist, etc.) or who prescribes the T itself (endocrinologist, etc.). Sometimes specialists have long waiting lists - especially if they specialise in treating trans* folk.
The other thing they mean is that their psychiatrist/therapist will not write them the approval letter for several months/a year.
As Hex said, it depends on who you see and where you live in the world.