As an American who plays a lot of British characters (just ... not in this game BUT I SWEAR, I REALLY DO), honestly a helpful thing to do is to immerse yourself in British culture.
Watch some British sitcoms, British drama, British talk shows, read some British books, listen to some British podcasts -- even just listening to British radio can help (Radio 1 or XFM are the ones I like, and they both offer livestreams).
Find a British actor you like and look around on YouTube for interviews with him on UK talk shows -- or even American talk shows, as then you can listen to the differences in syntax, etc. Even trolling YouTube for UK vlogs can be fruitful!
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Watch some British sitcoms, British drama, British talk shows, read some British books, listen to some British podcasts -- even just listening to British radio can help (Radio 1 or XFM are the ones I like, and they both offer livestreams).
Find a British actor you like and look around on YouTube for interviews with him on UK talk shows -- or even American talk shows, as then you can listen to the differences in syntax, etc. Even trolling YouTube for UK vlogs can be fruitful!