It definitely is YMMV. I'm no singer, but my experience has not been anywhere nearly as rapid.
I also think dosage is part of this. If you take a higher dose, the odds are more certain you won't have as long before your voice starts to change. Some guys who care more about preserving a singing voice start on a lower dose and are much more gradual in working up to a 'full' dose to help try to manage the voice change better. It doesn't guarantee anything, but it can help. And low dose means LOW dose (start with 1/4 to 1/3 of ultimate goal, build up, as opposed to the common "low" dose of 1/2 dose up to full dose within 6 weeks!
I don't understand why your options are 'get T now or in a few months from now,' because you can always get a prescription filled and get taught to inject & then do it for real a few weeks later - whether it is self-injecting or having someone else do it (a nurse or whomever). The endo himself need not be the one to give an injection, so if he is the one who is very booked, that shouldn't matter.