For my psych, privately in Melbourne, I had to wait around 3 months. Then I had approx 4 - 5 sessions, at least two weeks apart.
After that I had to wait another 3 weeks after signing a consent form with my administering GP.
This was for approval for testosterone therapy, for you it may be different if you are seeking feminising hormones.
This was quoted to me as being far quicker than going through public sources (such as the gender clinic in south-east melbourne).
So In my experience, I would strongly suggest that it is not worth travelling interstate, unless you are a millionaire.