This question comes up with some frequency in this sub, and I have answered it a few times. You can read a couple of those answers here, and here. The related myth about the American accent being "closer to the original British accent" or closer to Shakespeare or closer to Elizabethen English is addressed here.

To paraphrase again: there was likely a distinct American accent just as soon as there were enough American-born adults around for English/British visitors to converse with to notice. This could have happened as early as the late 1620s in Virginia and the early 1640s in Massachusetts, though the initial American-born populations were tiny. They were swamped by a much bigger influx of English immigrants shortly after the colonies were established. In Plymouth and Massachusetts, this coincided with the Great Puritan Migration that started with the Winthrop Fleet in 1630. Thus, it's likely an accent wouldn't have been noticed until about mid-century, since so many of the adults until then would still have been immigrants, and linguistic "leveling" was still in its early stages among their offspring.

According to historian Paul K. Longmore, by the 1670s, there were enough American-born English speakers that English-speaking visitors from England would have noticed a distinct American manner of speech. Historian and language scholar Allen Walker Read found evidence going back to 1663 of commentary by English visitors of how Americans were using words and language differently from how the language was being used back in England. From that point forward, there is regular indirect evidence that there was a separate American accent from any accent heard back in England.

The first direct evidence of an American accent comes from Hugh Jones, an English-born professor who taught at the College of William and Mary in Virginia between 1715 and 1721. After his return to Great Britain, he wrote a book called Accidence to the English Tongue, published in 1724, which described some of the new innovations of American speech (like pronouncing the letter s in some words closer to z as in "has" and "raspberry"). He also commented that the American accent was much more uniform and much more old-fashioned, which he considered a good thing since he thought the accents then developing in England were becoming unsophisticated.

From that point forward, there are regular mentions by English visitors noticing the American accent, or English immigrants sticking out as a foreigner. In fact, as early as 1739, Benjamin Franklin had written in Poor Richard's Almanack about regional differences from colony to colony in the American way of talking.

By the time of the American Revolution, not only was "the" American accent commented upon regularly, but regional and social dialects were being commented upon as well. These accents have not remained static, but have continued to take on new innovations in pronunciations, speech patterns, word choices, vowel changes, and linguistic drift since then. Accents develop and die out with some regularity. A notable example is the upper-class Northeastern speech used by Franklin Roosevelt and George Plimpton that was a prestige accent quite prevalent at the turn of the 20th century (and often mimicked by non-native speakers) but is now virtually gone. Another that appears to be changing dramatically if not dying off is the r-less Southern speech pattern of people like Jimmy Carter and James Carville, while other regional Southern dialect markers are supplanting them and expanding their reach.

You might want to consult this sub's FAQ as well, which provides some other answers to this question in addition to one of mine. (CTRL+F "accent" on that page and you'll find a list of them.) While not in the FAQ, you may also want to consult this post by /u/YuunofYork, who answers the question from more of a linguistic point of view rather than a historical point of view.