These men lived in a world dominated by Britain. They lived in a transplanted English society. Modern films seem to always depict our forefathers speaking in modern American accents but isn't it more likely that the men mentioned in the subject question spoke with a heavy British accent??
It might make more sense to rephrase your question a bit - what do you mean by 'a heavy British accent'? Pronunciation changes over time and has always varied regionally, so there is no single British accent today (and there wasn't in the 20th century). So, the founders of the US wouldn't have spoken with a British accent or an American accent as they exist today. Their speech would have been mostly influenced by where their families came from in the UK, so in that sense it would be a type of 'British accent', but of course those accents are what developed into modern American English.
If you're wondering which modern accent would sound closest to how George Washington spoke, you might want to ask in /r/linguistics (maybe the Q/A thread?) as there are probably some folks over there who are knowledgeable about the development of American English.
(not sure if this is an acceptable top level comment as I don't really know the answer to your question, just trying to point you in the right direction)