It varies. I say it took 3 months to get my HRT letter, 13 to get the initial, if too low, E-only prescription, 22 for minimally effective HRT (AA added, E too low for me,) and 34 for HRT that was right for me. By the last, I'd been fully transitioned for three months - I hadn't earlier because I needed the mental changes I got when the AA was added.
Transition felt impossibly far away for a long time...honestly, I don't think I believed it was real and within my grasp until less than a week before I transitioned at work - the moment the meetings with my coworkers were actually scheduled. I'd come up with so many timelines that my endo and therapist required so I could prove I was serious and diligent, but then had fall apart because of the lack of progress on HRT that I just couldn't believe it until it was effectively in-hand. So much of the time was spent trying to convince providers to get on with things. I do understand and agree that caution is necessary, but what I experienced went far beyond reasonable caution.