There is a lot of information about this in the Popular Questions (FAQ).

The FAQ has already been linked, but I'll give a brief answer in summation.

"The English accent" isn't an easily defined dialect. It isn't today, and it certainly wasn't in the past. Many differences in modern pronunciations can be traced back to earlier ones. Whether or not r is pronounced at the ends of words/syllables in many contexts (called rhoticity/non-rhoticity), for instance, was a point of difference in dialects of English in part centuries. And there are both rhotic and non-rhotic dialects in both the UK and US.

But if we define an arbitrarily far back common Early Modern English, it'd probably not sound like either. The not pronouncing r at the ends of words is the newer system. But many features in American English are relatively newer--American dialects have a group of innovations in the vowel system.

Here is a sample of what English in London would've sounded like in Shakespeare's day. Some features are more American, like rhoticity (pronouncing r at the ends of words). The vowel system is more like modern British English. Some features, like "film" being two syllables, are found in Hiberno-English (Irish English).

I think what you are referring to is the 'great vowel shift', which was the change in English pronunciation from old/middle English to modern English.

Broadly, the great vowel shift seems to have been London driven, and the further you get from London the less the vowels have shifted. So for example, a Scottish accent probably sounds more like middle English than a London RP accent.

Much of the initial emigration to America from England happened during this period when pronunciation was changing. American English branched off from British English as the settlers' accents developed in isolation of England. I think it would be a stretch to say American English sounds like Middle English, rather it is a different evolution from the same starting point.

Any American language experts on here who can say which American accents are old vs which are relatively new?

It's not very accurate. However one huge noticeable feature of English accents is non-rhoticity (silent or vowelized 'r's) and this happened in England after the colonization of America. Some American accents, like the Boston accent, also lack rhoticity as well as many English accents still having it, so it's by no means a universal rule. The accent changes a lot over time and in different places and social classes. Shakespeare is often compared to the West Country accent, with some slight shifts in vowels.