I posted the following elsewhere on Reddit recently, which should answer your question. The AskHistorians' FAQ attempts to answer the question, but really only dances around it.

The first thing to know, is that no matter when the first American accents were heard, those accents probably sounded little like any accent currently heard in the USA today. And same goes for the English (pre-UK) accents of that time. They wouldn't match directly with any current English accent. And because of the old fashioned way they would talk, they'd sound more like each other than either one would sound like an English or American accent of today. For example, the contraction "Twas" was in common use on both sides of the ocean at the time of the American Revolution, while it's almost entirely absent from any American or British dialect today.

Further, most people who pose this question seem to be under the impression that as long as the British controlled the colonies, then the people sounded British, and once the Brits left, then the change happened.

But that's not what happened, because that's not the beginning of what became the United States. Nine of the original Thirteen Colonies had Europeans living in them before 1650, and all but Georgia had European inhabitants before 1680. Going back to Jamestown, there are 167 years of colonists and colonial history before the Revolution began.

Many people who ask this question also assume that accents are relatively static. They are not. Accents become popular and fall out of fashion all the time. A recent example of this is the disappearance of the "Transatlantic" or "Mid-Atlantic" accent, that faux-English-sounding accent heard in movies before the end of World War II. It wasn't just movie stars like Cary Grant and Katharine Hepburn who used it, but it was the accent of Eleanor Roosevelt, Franklin Roosevelt, and Teddy Roosevelt, and many other families of the East Coast elite, including the so-called "Boston Brahmin", though that accent is slightly different. Some of the latest (perhaps last) celebrities to really have an upperclass East Coast accents were George Plimpton and William F. Buckley, both of whom passed away in just the last fifteen years. There's probably some old timers out there who still have it, but the accent is dwindling quickly.

Less well known is the "American brogue" that went out of fashion in the mid-19th Century. It's this type of accent that Abraham Lincoln evidently spoke with, and so did many others "out West", while a more "standard American" version of it was found in the middle class in most major northern cities and lasted into the early 20th Century. It was rarely heard among those born after the turn of the century, but luckily, it is preserved on many early recordings. The Hylan and Taft recordings may be the most distinct examples:

Capt. Chambers D. Reamer, Civil War veteran of the 106th Ohio Infantry, born 1843

Woodrow Wilson, President 1913-1921, born in Virginia, 1856, and raised throughout the South before moving to New Jersey

William Howard Taft, President 1909-1913, born and raised in metro Cincinnati, Ohio, 1857

John F. Hylan, mayor of New York City 1918-1925, born and raised in upstate New York, 1868 - relevant portion starts at 0:27

So, onto the question at hand: when did distinct American accents emerge?

Distinct American accents emerged no later than the early 1700s, and in all probability, they emerged with the first American-born generations back in the mid-1600s, though that can only be inferred indirectly due to lack of any direct mentions of accents in early surviving documents. Here's some of what we do have in the way of historical evidence:

A guy named Hector MacNeill was born in England in October 1728, and immigrated with his family to Boston when he was eight years old, in July 1737. He later wrote his memoirs in which he retold this anecdote that happened shortly after his arrival:

"A Little Lad who lived next door Observeing me a Stranger, fell into conversation with me, and being highly diverted with my manner of Pronounceation, (whither to amuse himself or some of his comrades to whom he intended to introduce me) led me out into the streets where we soon met with other Boys who were going to see a Ship Launched..."

So we can safely say that by the 1730s, Massachusetts English had an accent distinct from one heard back in England.

And we can go back further than that, with the case of none other than Benjamin Franklin. In 1773, he was caught up in the Hutchinson Letters Affair, a scandal involving the then-Governor of the Massachusetts colony and some leaked correspondence that Ben Franklin gave to the Boston Gazette. Long story short, there was a trial back in London, and during closing arguments, the prosecutor lambasted Franklin for the leak and called attention to his accent:

"After the mischiefs of this concealment had been left for five months to have their full operation, at length comes out a letter, which it is impossible to read without horror; expressive of the coolest and most deliberate malevolence.—My Lords, what poetic fiction only had penned for the breast of a cruel African, Dr. Franklin has realized, and transcribed from his own. His too is the language of a Zanga:

" Know then 'twas — I.

" I forg'd the letter — I dispos'd the picture—

" I hated, I despis'd, and I destroy."

This is important for a couple of reasons: First, Ben Franklin was born in Boston in 1706, and second, he was the son of an English immigrant. Thus, we can conclude that there was a distinct colonial accent at least as early as his Boston childhood in the 1710s, and very probably earlier than that. It also gives us insight into the difference in speech between the two countries--the Americans were dropping their "e"s on most past tense verbs, like most accents on both sides of the pond do today, but the English had not yet made that change.

In New York as in Massachusetts, distinct English accents began developing in its early decades. In 1664, the English took occupation of New Amsterdam from the Dutch, and while Dutch continued to be a first language for a majority of New York's population for decades to come, many of them began to learn English as a second language.

This is exemplified in a surviving letter from 1668, written by New York colonist Jeremias Van Rensselaer and addressed to relatives back home in the Netherlands, and translated from the original Dutch:

"Now it seems that it has pleased the Lord [to ordain] that we must learn English. The worst of it all is that we have already for nearly four years been under this jurisdiction and that as yet I have learned so little. The reason is that one has no liking for it."

But learn it, they did, or, at least, their children did. And with it, another uniquely American accent emerged--the Dutch-American accent of bilingual Dutch-Americans when speaking English. This can be seen in the phonetic spelling of the half-literate real estate tycoon Catherine Brett. Born in Manhattan in 1687, she once wrote a letter that started off:

"Sir:

"Afther my Kind Respacks tou your selfs en T yours this Comes tou Retorn you thancks for your Favers tou your spous, Sr. En I had a Litel descors about her desorder. Sie has bin aflectid weht wyle sie had that Dimnis on her syht en accorden tou wat axpirens I haff had. I thack it that the esstirrix is the prinsibel cors. Thes Drops at vary goud tou supres the vapers. My Dater has Resifd grat Rilyffe by thacken of them..."

With revealing spellings like "retorn", "desorder", and "aflectid" for "return", "disorder", and "afflicted", we can infer from her phonetic spelling how her English-American contemporaries might have sounded like to her. Her letter also reveals a Dutch influence over some of her words: "sie" for "she" and "haff" for "have", for example. And even though this isn't the "American" accent you're probably thinking about, it gives evidence of a distinct accent used by an American-born native English speaker, even if English was her second language.

(...to be continued...)