After registering, how long can I expect to wait for an appointment

One week

Within one week.

My mother's appointment was set for under a week.

In and out was under 1/2 hr

I am not clear on your question. How did you register? With whom? I called the number provided by the Health Department and was scheduled for the following Tuesday, less than a week after I called. I was told that I had the last available appointment, so it may be the luck of the draw. I registered online AFTER receiving an appointment, per the confirmation email I received.

I didn't reg for first shot. I went to the first come, first served thing in Jan. They gave me card with appt already scheduled for second shot at that time.