I can only give you info from my family, but I know it depends a bit on how it gets offered. The current pattern seems to be a letter with a link inside, that link offers local sites and the site page lets you book a date up until a about a week into the future. The reason the sites are looking like that is because the vaccine deliveries have been a bit inconsistent, so they wait until delivery dates are confirmed. The GPs are phoning/sending texts as well - but again they've only got visibility a week or so ahead.
Unless you actually tell them that you don't want it (or your record hasn't been updated to show you having had it at work or something) they just seem to try again a week later with another text, and ask you to have another go at making a booking.
By May, I think things will have calmed down a lot, and a lot of sites will have AZ sitting in the fridge so they can be confident they'll have a supply for a couple of weeks ahead, and booking systems to match. In other words I don't think there will be a problem for you (* this comment comes with no warranties, or guarantees)