So there's a lot to get into here, but I'll try to keep it brief. I have a degree in Linguistics and I feel fairly qualified to actually answer this.

So language is always changing, right? The language that was spoken by the first British colonists sounded nothing like the language spoken today on either side of the Atlantic to our modern ears, but it was undeniably English. They spoke Early Modern English, various dialects which have influenced our Modern English dialects, but none of which exist precisely as they did in that time.

So let's rephrase this question in a way that we can actually Answer.

A) what makes someone sound "American" vs "English", and

B) where and when did that divergence take place?

So with the caveat here that it is almost impossible to make a broad, sweeping statement about dialects that is true either synchronically (of various dialects at the same time) or diachronically (of the same dialect over the course of time). Every single person has a unique idiolect, so language evolves by the process of being analyzed and reproduced by individual speakers. So with all of that said, let’s try to make some broad, sweeping statements. I will attempt to describe some features which distinguish mainstream American dialects from mainstream UK dialects, and pinpoint where they diverged.

Rhoticity - Probably the most obvious distinction between North American English (Most dialects spoken in the US and Canada) and the rest of the English speaking world is rhoticity - the retention or loss of postvocalic /r/. That’s a lot of jargon, but it basically amounts to deleting /r/ at the end of a syllable if it is not immediately followed by another vowel. So where an American or Scot - or nearly any English speaker 500 years ago for that matter whose accent is rhotic - would say car as /kar/; but someone whose accent is non-rhotic - who drops the /r/ at the end of a syllable - would say /ka:/ or “kah”. However, all of them would pronounce the sentence “The car arrived” more or less the same: /ðə kɑr əraɪvd/.

The origins of nonrhoticity are fairly well studied. It goes back to around the 15th century in Southeast England, and can be seen first in vocabulary. English contains many etymological doublets which show the history of a word as it was borrowed in multiple forms in English. For example, captain, chief, and chef are all the same root word, borrowed into English from Latin, Old French, and Modern French respectively, but having three distinct meanings in today’s English. This can also happen between different dialects of what are considered the same language today, and this is the origin of many common English words like ass or bass (from ME ars and bars; the former is retained outside of North America as arse). This deletion began before other coronal consonants like /s/, /z/, /t/, /d/, /n/. These /r/-less spellings become more and more common over the course of the 16th and 17th century, but this process was incomplete, and many people today still don’t drop their /r/s, even in England. Furthermore, many American dialects adopted this particular affectation later on. As with any linguistic change, this progress was noted even in contemporary sources. Ben Jonson’s posthumous 1640 work English Grammar says:

[the sound /r/] sounded firme in the beginning of words, and more liquid in the middle, and ends.

The majority of American settlers came from rhotic areas, at a time when the language was in flux. As r-lessness spread throughout southeast England, it came to be accepted even in polite and high-class speech, and after the War of Independence, r-lessness became seen in some contexts as a sign of prestige, of association with the mother country. This association is often thought to be behind the r-lessness of traditional southern dialects. Think Chahlston drawl, or Foghorn Leghorn. The theory is that Southern Aristocrats were more likely to have their children sent back to England to be educated, and since someone’s dialect is formed by their peers in their youth, that had a long-term effect on the aristocratic dialect in the south, which in turn had a trickle-down effect on various other southern dialects, including the precursors of African American English. A similar effect might be behind the loss of /r/ in New York and New England as well, but this was considered lower-class in these areas because it was imported by merchants rather than aristocrats. Neither ever spread much beyond the east coast because that was where wealth and prestige was concentrated in the early days of the American experiment, as well as being where anyone who had contact with the mother country lived. The r-less pronunciation being seen as prestigious can be seen as late as the 20th century in the emergence of the tinny Mid-Atlantic accent among people like the Roosevelts and Hepburns. This is a constructed accent which was created to make actors be able to sound "neutral".

You can actually look at American Dialect Areas and see the history of settlement. New England was heavily influenced by England, and this shows in their modern R-less dialects. Meanwhile, further south, Pennsylvania and upstate New York were settled by Quakers, whose dialect was rhotic. The coastal south I have already touched on, but the inland south was mostly settled by poor scots-irish immigrants, and their language was the precursor of modern Appalachian English; though decades of isolation have led it to morph greatly from its origins, its roots can be seen in its unique vocabulary, things like retaining “yonder”.

Here is a map of American dialect areas from UPenn. A lot of these features emerged after the American divergence from the other dialect areas, so I’m focusing on the most salient differences between AmE and BrE, rather than focusing on the differences within AmE. Still, here’s this for reference.

There are numerous other isoglosses which distinguish Americans from Brits, many of which involve vocabulary, but we’re focusing on pronunciation. I have focused on one particularly salient feature as a way to explain how historical linguistics is examined. I hope that was helpful.