I got my blood tests done before getting my T letter, since I didn't want a delay. My therapist faxed the letter to my physician, and that same day I got a physical, signed a consent form, and got my prescription. Because of his schedule, I couldn't see my physician and get my injection until two weeks later, but if his schedule had been free it could have been almost immediately after the T letter.
I went to 3-4 appointments with him before that, he had me read the standards of care and a few studies on the effects of testosterone, talked to my parents about it, answered questions, gave me a courtesy letter, did some quick psych evaluations, and started taking my blood pressure and weight. Itt depends on the doctor, but I don't see why you couldn't get T right after your T letter.