The branches that Lysander and Demetrius fight with are swapped later in the fight.
After calling Hermia a "dwarf", Lysander has a lot less mud on his face in the next shot.
The medal and the red ribbon that Bottom is wearing when he leaves the theatre are different from those he's wearing when he arrives at home.
In the play within a play scene, Bottom alternates between wearing and not wearing leggings.
The opening text tells us that the movie is set at "the turn of the 19th century," which would be around 1800. It meant to say "the turn of the 20th century," as the movie is clearly set around 1900.
Hermia and Helena pointedly insult each other's height difference ('dwarfish', 'painted maypole'), but Anna Friel and Calista Flockhart are only a quarter-inch apart in height.
The immortal King of the Fairies, Oberon (Rupert Everett), sports a very noticeable late 20th century vaccination scar on his upper left arm.