Pardon the phrasing of this question.
While we all know that the average American sounded indistinguishable from his British counterpart at one point, and the two deviated after their countries were separated, Americans "lost" the "British" accent at different rates depending on the region of the country he was in, and the social stratum he occupied.
I'm asking at around what time the Presidents started sounding 'more American' because that might give a more precise date, and because the President is a symbol of the country and its people in some ways, and the answer to this question would be helpful in identifying some overall, national trends.
While we all know that the average American sounded indistinguishable from his British counterpart at one point, and the two deviated after their countries were separated,
Actually, the two deviated much earlier. The question of the origin of American accents comes up really frequently around here - this thread has probably my favorite response by /u/lord_mayor_of_reddit : https://www.reddit.com/r/AskHistorians/comments/5toz0o/how_and_when_did_the_american_accent_come_to_be/
So to directly answer your question, the first President with an American accent would have been George Washington.