I can't answer your question about Jamaican accents, but it should be pointed out that the setup to the question is a myth that has been debunked, and written about in this sub before here, and here.

TL;DR: American English isn't closer to "traditional English". The confusion is because that most (but not all) American accents remain rhotic while most (but not all) British accents do not. But American English has diverged in many other significant ways that British English has not such as American English's tendency toward t-flapping. So with a word like "later", a Brit might not pronounce the "r", and the American would pronounce the "t" as a "d", but most British English accents of the 16th and 17th Century would not have done either, and they both would sound wrong and foreign to their ears, neither of them sounding particularly closer to the way they sounded.

As linguist Michael Montgomery wrote (emphasis mine):

"Americans, academicians and laypeople alike, have had a long interest in the history and diversity of their speech, evidence of a desire to trace their ancestors and to establish their roots. This interest has often manifested itself in less than objective ways, as in the wistful seeking of 'Elizabethan' or 'Shakespearean' elements in the English of the Southern mountains or elsewhere."

A more thorough explanation is provided in those previous answers.