When my (almost) 14-year-old son comes to visit, he does a mock accent "Pardon me, do you have any Grey Poupon?" It gets me laughing. But the girls over here love his American accent, so I don't see him disposing of it anytime soon. I'm hoping he comes to lives with us full time, so this chapter is still being written.
I do notice older folks on TV who have grown up in different countries. I was surprised to hear Gillian Andersen recently interviewed (by Graham Norton)...she grew up in England and when she's here, she explained, she folds back into the English accent, being surrounded by it.
My husband and I run a computer training company. We frequently get students in for a week at a time. I was suprised to discover once, on the last day, that we actually had an American! He moved over here when he was two, but his accent was pure English.
I dunno, I think it would be cool. But the cockney definitely would have to go!