I’d always assumed that Americans used to have 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. That’s not too surprising.

What’s 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. (Source)