Some people never lose their accent regardless of their time in exile, but English and American accents are not that far off from each other in relation to say Hungarian/Magyar vs American. Its very difficult to iron out your Hungarian accent into a perfectly sound American one due to considerable lingustic differences.
What are you talking about have you not seen any British films they have 500 dialects some doesn't even sound remotely English. You've clearly never heard Adele or Saorsie Ronan talk