I’ve noticed in films set in the early days of the USA, that all the characters have distinctively American accents even though many of them may have been only first or second generation migrants. How quickly was what we understand as an American accent today established?
/u/lord_mayor_of_reddit has previously answered How and when did the "American" accent come to be?
It is one of the many answers available in the subreddit FAQ about the history of English.
For more about the way American television characters talk, see the answers by /u/Dont_Do_Drama and /u/orincoro in response to Why do actors talk so oddly in American 1950s sitcoms? where one of the mods links to another answer by lord_mayor_of_reddit.
EDIT: Who do you hear in your mind when you say "American accent"? I'm curious.