American English wasn’t a thing until the Medieval period had long passed
Nor was modern British English. One of the defining features of modern British English is the lack of rhoticity (dropping the “r” sound), but that’s very modern, only happening in the 19th century. They have managed to recreate how English sounded in Shakespeare’s time by looking at words that were supposed to rhyme, and their meter. To me, it sounds like “pirate English”.
Whether modern American English is closer to Shakespeare’s English is a matter for debate. I’d say it’s closer than RP, but not as close as some rural British accents.