I grew up in London...and what I did after I came to the US is "shadowing." My Japanese friend told me about this. I saw thousand of movies and CNN news (thought very neutral American English) and just repeat what they are saying, maybe 1 second behind! It helped. So...I guess, like others suggested, you can see films (Don't see "Billy Elliot" or "Full Monty" or "Backbeat": Strong accent and dialect

) and BBC news on telly if you can get that.
And...yes, it is pretty much about intonation, and less strong vowels. And...not "buller," but "butter"

Speak slowly. You will be fine.