Hi there, I'm not a historian, but I hold degrees in European languages and as such I've a background in linguistics and as such I hope I can do this answer justice and hold up to the rigor of this particular sub. That said, I feel this question may well be better answered by those over in r/linguistics, unless there are any historians in the sub who specialise in language history.

Essentially, the short answer is no, not at all, but this myth seems ever-present for whatever reason, so there's no shame in being lead to believe such a thing. The long answer is of course a little more complex, but still a firm no. Before even getting into any history, one has to understand that there is no "original" accent for British English, in much the same way as there is no standard accent for everyone who lives in North America. Accents can often change over here from one town to the next, as well as each individual having their own affectations that affect their speech.

As well as this, language and accents change over time, as your question suggests, but this fact is true both in the UK and in the US/CA. When settlers from the UK and Europe left for America, they took with them whatever accent they held at the time. Often, you will find that groups of people from areas of the UK will have made the crossing at the same time, choosing to stick together once in America to start their new lives, as well as successive journeys made by friends/relatives to join them at a later date. This may have caused the basis for a particular area of America being lingustically distinct from another area, but one also has to consider that not all immigrants to NA were from the UK, and as such, linguistic quirks may have come about as a result of this mixing of cultures.

Once settled, accents in a given area will have begun to homogonise, as an area solidifies its identity, but then these accents will have developed in their own way, much in the same way as "back home" in the UK, changing into something altogether different to whatever accent they held when the original settlers arrived. Whilst an accent may hold some similarities with accents from back in the UK (such as some southern US accents having a striking similarity to some west country dialects), the hundreds of years that have passed between the settling of the Americas by British settlers and now, means that dialects and accents have had plenty of time to change on their own.

In short, just because settlers will have been isolsted from much UK linguistic influence does not mean that their accents simply stagnated and remained "pure". Differing accents will have merged with one another, or perhaps fell out of favour altogether in favour of another more prominent one, and all of these accents will have seen some level of change since the settling of NA by Europeans.

Thank you so much for your detailed reply!!