The 60+ have been able to book a vaccination since last week, I have booked mine and several others I know have also done so. (I think this has come up in several other threads, by the way).
To book you need your NHS number and you go to the central NHS site. Ignore the information on the front page about being 64 etc., put your NHS number in and see if it lets you book. If it does, you are fine to go ahead, if it doesn't it will tell you that you aren't eligible yet. You don't need a letter or text to do this, so long as you are over the 60 years.
However, this site is the central NHS one and therefore the centres you will be offered are the national ones, they'll all be in nearby towns/cities and will require some travelling.
If you want one of the local centres in your next door market town/GP surgery or whatever, this won't be offered if you book independently. If you want that, wait for your local letter/text or whatever, which might offer a booking via other means (eg. Swiftqueue) and which will give you different vaccination centres to pick from, much closer to home.
Once your booking is made, you are given a booking reference number which you take with you to your appointment. If people are turning up without a reference number and no booking, they will, of course, get turned away.