This question gets asked frequently on this sub and I have answered it at length before. If you don't want to read that whole wall of text, then I'll just copy-and-paste that post's TL;DR:

Circumstantially, the first American accents were heard just as soon as there were American-born children old enough to talk in 1610s Jamestown and 1620s Plymouth. Around 1670, local American accents such as these began their spread to being regional, used across hundreds of miles in various cities and colonies. There is convincing indirect evidence of accents between the 1680s and 1710s, while the first direct mention of an American accent comes from 1724 in Williamsburg, Virginia. By 1760, the American accent had been regularly commented upon, often remarking on different dialects heard in New England, the Middle Colonies, and Virginia/the South.

Some answers to this question often say something about the English spoken in Appalachia, or the Tidewater region/Tangier Island, Virginia, being close to how Elizabethen English used to sound, but that's a myth. In fact, by the time of the Revolutionary War, the Tidewater accent had already been mentioned as being a separate accent from the more general accent heard in the southern colonies.

Sources on all this along with a more thorough explanation can be found at the link at the top of this post.