Accents develop over time. What this means is that when people who speak the same language are separated, their accents diverge away from each other and you get regional accents which are distinct. I’m assuming you’re American, so think how different the accents in Boston, Minnesota, New Orleans, Mississippi and Los Angeles are from each other.

The UK today has a similar situation but even more pronounced, so that even small towns close to each other might have different accents (though this phenomenon is reducing since there’s less separation between people than in the past). Examples might be the Essex accent, compared with the Birmingham accent, the Welsh Valleys accent, the Manchester Accent, the Bristol accent, the Yorkshire accent, the Newcastle Accent, and the Glasgow accent. None of these sound like what you imagine the “Posh British” accent to be, but they’re all just as British. This was true in the 17th Century too - in fact truer since regional variations were stronger.

So the first point is that there isn’t one American accent, and there isn’t one British accent either.

Secondly, going back to the point about accents diverging, clearly the pilgrims who left Britain for America would have had the same accents as their fellow Brits when they left. But those accents (a) would have varied according to where each pilgrim came from in Britain and (b) would not be identical to accents heard today, though there would be similarities for sure.

Since then, Americans and Brits have been divided by the Atlantic Ocean, so their accents will have been diverging from each other for the last few hundred years. This means that inevitably they will both have changed to move away from the common accent they might have had back then.

So the ultimate answer to your question is that neither the current British accent nor the current American accent is the same as how people spoke when they started diverging, but clearly they each come from the same source - how British people spoke back in the 17th Century. But British people spoke in all kinds of accents back then so there isn’t really one common accent you can point to.