And an optional follow up question: going back further, what would Shakespeare have sounded like?
That should help. It's an oft-asked question, and it seems like this answer is best:
The primary American accent has changed less over the past 400 years than the primary English accent. The main version of the English accent spoken at that time sounded very similar to the modern-day American accent - with flat vowels, and hard ("rhotic") r's. In fact, there's an island in Virginia where the locals are believed to be speaking English very similar to the way the original colonists spoke. What happened was that accents in England changed over the past few centuries. Not all accents, just the ones spoken in the south of England, including London (the ones that became Received Pronunciation). The vowels changed, the r's softened. This change did not happen in North America. Hence the difference.
Now that's not 500 years back, but it should give you a pretty decent idea for 400 :).
You may be interested in the 'American and British accents' section of the FAQ.
The idea of a British accent is problematic - there are many, and they vary a great deal. Non-Brits often seem to think that we all speak like the royal family, or Dick Van Dyke in Mary Poppins. (I'm from London, and Americans often think I'm Australian.)
In fact, the variation between, say, Cornwall, Birmingham, Newcastle, Belfast, Glasgow and the Highlands is huge. (There are some accents, especially in the North East, Lowland Scotland and Northern Ireland that I have trouble understanding.)
It's likely that before nationally broadcast media accents were even more distinct.
Different parts of America were settled by different populations of British settlers. Would it be possible to identify the influence of various British accents in the respective areas of the US?
p.s. Apologies to the OP if I seem to be derailing your thread.