It is obvious to assume that Americans used to have accents similar to today’s British accents, and that American accents diverged after the Revolutionary War, while British accents remained more or less the same.
Americans in 1776 did have British accents in that American accents and British accents hadn’t yet diverged – an answer which is not too surprising.
What is surprising, though, is that those accents were much closer to today’s American accents than to today’s British accents. While both have changed over time, it’s actually British accents that have changed much more drastically since then.
First, let’s be clear: the terms British accent and American accent are oversimplifications; there were, and still are, innumerable constantly-evolving regional British and American accents. What most Americans think of as the British accent is the standardized Received Pronunciation, also known as ‘BBC English’.
While there are many differences between today’s British accents and today’s American accents, perhaps the most noticeable difference is rhotacism. While most American accents are rhotic, the standard British accent is non-rhotic. (Rhotic speakers pronounce the ‘R’ sound in the word ‘hard’. Non-rhotic speakers do not.)
In 1776, both American accents and British accents were largely rhotic. It was around this time that non-rhotic speech took off in southern England, especially among the upper class. This prestige non-rhotic speech was standardized, and has been spreading in Britain ever since. Most American accents, however, remained rhotic. However, there are a few fascinating exceptions: like Irish and Scottish accents, New York and Boston accents became non-rhotic.