To the average American people from the UK have two accents English (England) and Scottish. I too fell into the trap of thinking that all British people sounded pretty much the same. I mean how could they sound different the whole country is the same size as most states.
In the USA I never really noticed to much of a difference in regional accent. I basically heard 3 or 4 different accents. The West coast, Midwest, Deep South and the East Coast Mostly North East coast.
In England that is completely different. Not only are the accents based on the region you come from but they also use different words. For example based upon where you live if you want a sandwich you would ask for a sarnie, a barm, a butty, a sandwich, or a bap it just depends on where you live.
The accent also changes from one region to the next. In order to help you out on your next visit to the UK as to what to expect from where I thought I would share this regional accent map. Hope it helps :) It is so fun to listen to the difference!