Mis-stated question. Instead: When did american accents begin to develop? From the very beginning.
The american east coast was formed by four different groups of immigrants from different areas of Britain who already spoke with different accents. (To us they would all sound much more gaelic than british do today.) These groups spread in horizontal bands across the united states, and the cultural horizontal ‘bands’ in the country reflect the westward expansion of those early settlers, and how they carried their languages with them.
The intentional ‘middle atlantic’ accent was something you learned, just like received pronunciation in the UK until the underclass revolution of the 1960’s, the marxists, postmodernists, attempted to undermine all western aristocratic values.
Universities attempted to quash dialects for the competitive marketability of their students, and radio, then television assisted in the homogenization of the ‘Indiana’ pronunciation throughout most of the country exclusive of the lower classes. Most of this dramatic homogenization has come about since 1980.
Our accents may not sound as distinct to others as do those of different regions of the UK, particularly in the underclasses. But in the states, your vocabulary, body language, and pronunciation are your primary forms of status signaling, and we can tell, most of the time, at least which region if not which state or city each of us is from.
The most interesting property of american pronunciation is probably the least discussed, and least well known, which is that the majority of white americans are of germanic rather than anglo extraction. And so the american speech pattern inherited german monotonality rather than british and gaelic tonal accents.
So Americans speak the vocabulary and grammar of the english language with rather dry german pronunciation so to speak. If you hear English in the Gaelic or the Old English, it’s more melodic. There is a tempo to it. It’s more expressive.
When I teach people from melodic backgrounds how to speak english (particularly Indians), I tell them to practice: speak like a robot-voice in the same tone, deep in your chest, with continuous air, and beat your chest every syllable at a constant rate – and while it sounds silly, this technique will teach you the proper pace of english speech.
If you look at this map, you’ll see the westward migration of the dialects as we spread westward.
( PS: As an aside, the actor who Portrays John Adams was chosen, as is common in Hollywood representation of the Founders, as a means of insulting the great man. He had more in common with a Field Marshal than he did that wimpy little fellow. Founders were tough, hardened, empirical people. On a scale we cannot imagine today, because no one like them exists today. )