The sound of Washington's voice specifically will likely never be known, obviously. Most sources I've read on Washington have placed more emphasis on his sheer presence and sometimes even his lack of speaking when it came to how he was like in person.

But what we do know is that what we think as the posh and elegant British accent of today is actually a fairly recent development. Original colonial settlers of British origin had a rhotic accent (basically pronouncing your R's) and were overall much closer sounding to modern American accents than the British now are. When British influence ended, however, both accents were left off to develop and change on their own. They were essentially branched off each other as the British started to speak non-rhotic and Americans continued speaking rhotic (generally) as they both do today.

But of course, regionality throws a bit of flair on accents both then and now, so a Virginian surely had their own way of speaking compared to a New Englander. I personally do not know the specific extents of the larger differences (if any) or how they actually sound. But it's safe to say neither Washington nor most of the people in colonial America at the time sounded much like what we are led to believe in the majority of TV/Film accounts.

I wouldn't quite say he spoke like a hillbilly southerner, but given the choices between British and hillbilly, he was most likely closer to the latter. It's a fairly interesting topic overall and I believe there have been more general questions posted about colonial accents in the past that may be worth taking a look at.