It really differs. The usual scenario is to be in therapy for at least 3 months before your therapist is willing to write the letter. After that, it will usually take some time before an endo will see you.
I happen to be very lucky. My therapist is trans herself and is very well known for her work in the trans community. Therefore, she is more open minded and flexible about the letter. Instead of following a time frame, she judges hrt letters based on the level of readiness and whether it would be appropriate for her patients. It is a case by case scenario. Some will get it soon, some latter, and others she may not feel comfortable writing the letter. In my case, she felt it was appropriate to write one very early. After she approved me, I was able to get an appointment with an endo a few days latter (happens on Monday). This is all very rare. The other endo she recommended doesn't have any openings until a few months from now.
So everything really depends on the flexibility of your therapist and how soon you can get an appointment with an endo.
Also, you can start hormones without therapy through informed consent. But I really would recommend going through your therapist. It's not about doubts or anything like that. I just believe having someone on your side to help guide you through the changes is really helpful.
