I'm in the US. "Shitshow" doesn't even begin to cover it. The UK is going to blast past us with the percentage of population vaccinated in weeks, and I am DELIGHTED for you -- I can't help but be happy every time I see one of my friends back home has had theirs.
In my state, there are 90 vaccine sites. There are 70 separate websites you can check for appointments at these sites. There is a main "state system" which is connected to maybe a dozen at best, and which isn't working very well emails not being sent, etc. And this system is definitely for the computer-savvy only. There is a phone number for people who don't have computer access; hold times are hours long, people get disconnected, and there are no appointments at the end all it is, at the moment, is someone entering your information into the "state system" which, as I said, doesn't work and isn't connected to most of the sites.
There are 4 million eligible to receive the vaccine right now, and circa 200k doses getting to us each week, to be used for first and second doses. Appointments are rare, and go as fast as they appear. Waitlists are thousands of people long.
There are six "megasites". They ran out of doses, as did the local hospitals, despite the state allegedly getting nearly double the initial predicted supplies. DH had to have his appointment rescheduled. When he did get one, he did end up queuing for three hours, but he got it.
So I am absolutely thrilled that this rollout isn't an omnishambles everywhere. You're very fortunate!