Depends on a lot of factors, but a gender therapist will typically be willing to write the necessary letters after just a couple sessions, mine pointed me towards the endocrinologist at the second session. The endocrinologist on the other hand can be booked solid. Transgender hormone therapy is a specialist field with not too many doctors in it, mine originally scheduled my appointment four months from when I had gotten all the papers in order. They called and rescheduled for next week though as someone cancelled I guess.
I've heard in some places the wait list can be a year long or more, it really depends, that said, it doesn't have to be an endocrinologist specializing in transgender medicine, but their additional experience will serve you better in the long run. There doesn't really seem to be a consensus on how to do hormone therapy for transgender people, I've only looked at MTF, but so much of it is contested, like whether or not to prescribe a progesterone with the estrogen, if anti androgens should be stopped after a few months, or continued until the boys are physically removed, etc.