I've just booked mine for next week and some time in July at a local vaccination centre. There was a queue of a minute or two to get onto the website, but then it went smoothly.
I got a confirmation screen with two reference numbers on, which I took a photo of, and asked for details to be sent by text, which I haven't received, but I assume they'll come through in time.
DPs just had his second as he's got a health condition, so if vaccination passports become a thing, it means we'll be able to go on holiday or do whatever else rests on being vaccinated, some time after my second jab.