Just to share how it works in benevolent dictatorships (Singapore), registered interest in appointment about a week ago. Received an invite to make an appointment today, clicked on the link and filled in a form and then made my appointment at the nearby community centre for 8 April. and second 3 weeks later (Pfizer it would seem, the other one being used here is Moderna, which has 28 day interval). My partner did same (we're both around 50yo +/-). The process took about 2 minutes. High density populations, and a technology oriented government seem to make things a bit easier. Although for 60+ they've been sending out letters, and allowing phone appointments, but will be all electronic for younger generations.
Meanwhile back in regional Victoria, my 75 yo mother is not quite there yet with appointment has rung several clinics etc, her GP maybe getting it during April and taking appointments then.