Is there any way of tracking or knowing what the various British accents/dialects sounded like in the past? Have they changed appreciably? Has American accents/dialects changed much? When did the break between English English and American English happen? How much do they both differ from their roots?
There are various ways. Studying rhymes from old poems and songs is a big part of it. Spelling variants can say a lot, as English was more phonetic before spellings were standardized. There are also several sources in which contemporaries described the sound.
We have very clear contemporary documentation that Shakespeare and his company (unlike most modern Englishmen) had rhotic accents. There are texts describing their R's as a rather "doggy sound," almost a growl. The Original Pronunciation of Shakespeare was far more similar to the stereotypical "Pirate accent" than it is to Received Pronunciation.
It is important to remember that the is not one British accent and one American accent. There differences within each category are greater than the differences between their respective means. This can make it hard to identify when the accents diverged.
There is a wider variety of regional accents in Britain than there are in America. Regional accents in America show clear connections to those of the regions from which the bulk of the early settlers emigrated, but different groups coming from different places at different times eventually had their accents merge together
Today most English accents drop their Rs, and most American accents do not. There are still some exceptions, but fewer than there were in the past. We are still familiar with the non-rhotic Boston accent, but the non-rhotic southern acrolect (like what my father's mother used) is dying out.
It is generally believed that American accents tend to be somewhat more conservative than British ones, and that rural accents tend to change more slowly than urban ones.
It has been claimed that the inhabitants of Appalachia sound just like Shakespeare would have, but that has been pretty much debunked. Their accent is one of the more conservative ones, but it was introduced by the Scotch-Irish/Ulster Scots rather than the English and has still some changed over time.
Around the time of he American Revolution, men like Noah Webster decried some of the changes of pronunciation that were just coming into vogue in the cities. He hated how city folk were starting to insert a consonantal i sound before the letter u, making the letter sound like the world "you."
Going even further back, there are recordings of Theodore Roosevelt, Woodrow Wilson, and other prominent American politicians on youtube from the turn of the century. Put them up against the equivalent early 20th century British politicians and they sound remarkably similar (at least by comparison to today's accents).
You are describing the Mid-Atlantic accent, a standardized from of pronunciation that was purposefully designed to lack both clear Americanisms and Briticisms. It was sort of American version of Received Pronunciation developed in New England. Elite boarding schools used to train their pupils talk this way, as did many acting schools. This contrived acrolect does not tell us much about how the common man spoke.
We do however have many recordings from the late 19th century, when a concerted effort was made (by the Census Bureau, if I remember correctly) to document how the common man spoke. I recall someone noting that these recordings show absolutely no difference between the accents of blacks and whites; former slaves used the same accents as their former masters. We cannot really be sure that those recorded spoke the way they normally would though; they may have tried to sound more proper since they knew it was being preserved for posterity.
