In the 1590s, Wessex owns "tobacco plantations in America". There were neither tobacco plantations nor English colonies in America in the 1590s. The Roanoke colony at North Carolina (called Virginia at the time) failed in 1587, and tobacco monoculture did not begin in Virginia until after 1607. The filmmakers knew this.
Viola hears the cock crow and refers to it as a rooster. The word rooster is not recorded before 1772 and is said to be an American coinage to avoid the associations of the older word. Even today it is rarely used in the UK.
William Shakespeare/Romeo tends to Ned/Mercutio by kneeling to Mercutio's right, and, in doing so, violates the first "rule" of stage acting, which is to never hinder the audience's view of the stage or the actors.
One of the main plots involves William Shakespeare creating the story of Romeo & Juliet, making it up as he goes along. In fact, these characters and their basic story were written about before Shakespeare was born, and he was adapting the older tale for the stage. The filmmakers knew this.
Near the end the Queen commands Wessex to pay off the wager of 50 pounds which Viola carries in a pouch. At that time, a British pound coin would have contained one troy pound of silver (hence its name). A troy pound weighs approximately 75% of a normal pound, and would therefore mean the sack she is carrying would be very heavy, much heavier than than it appears to be in the movie.
The sound of the tambourine, played by William Shakespeare with the musicians at the dance where he meets Viola, continues after he has stopped playing. However, given that Will was not originally part of the music group, it is reasonable to assume that there is a second tambourinist somewhere.
William Shakespeare's Romeo & Juliet is generally thought to have been performed for the first time in 1595 or 1596. Shakespeare in Love (1998) has it performed in 1593, but this is part of the "secret history" which the film is "discovering".
When Will and Viola are in bed together the morning after their first tryst, the bed sheets are pulled to provide concealment in such a way that only someone not in the bed (and off camera) could physically do.
Just after Lady Viola tears off her wig, accidentally revealing herself to be a woman, Sam (the actor playing Juliet) is shown standing next to her, his dress hanging down normally. However, in the next shot of Sam, his skirt is up around his shoulders, back where the Master of the Revels had raised it a few minutes earlier, erroneously believing Sam to be the woman in disguise.