Get in touch with your local Council at your new address - Electoral Roll dept to get on the list where you are. This even applies if you move within the same borough/area. They will advise.
But I think you will not go on the roll until next round of applications which most do every 12 months
No, no record of where you were before. A form comes to the house every year for us to fill in who lives here now. If you contact the council offices they will tell you how to register the change before then, if you want to.