Short, somewhat misleading answer: no.
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.
Then why does everything old and/or foreign have to be spoken in a British accent? I’m looking at you, The Last Station.
(via The Hairpin)