You know what really blew my mind? The closest accent to the one in Shakespeare's time is not modern British English, it is the American accent.
Apparently what happened is that after colonizing North America, the British started developing a different accent which over many years has led to a huge departure from what it sounded like in Shakespeare's time.
The British immigrants in the US didn't follow the trend in England and kept their original accent and therefor the US accent is now more similar to how English was spoken way back in Shakespeare's time.
TL:DR
If you want to be time period appropriate, you should always use and American accent when reciting Shakespeare.
Apparently what happened is that after colonizing North America, the British started developing a different accent which over many years has led to a huge departure from what it sounded like in Shakespeare's time.
The British immigrants in the US didn't follow the trend in England and kept their original accent and therefor the US accent is now more similar to how English was spoken way back in Shakespeare's time.
TL:DR
If you want to be time period appropriate, you should always use and American accent when reciting Shakespeare.