This is a meta movie, so it's not just about Shakespeare writing a play full of his usual tropes, it also contains Shakespeare's usual tropes – comedy, tragedy, cross-dressing, and a nurse.
Like a nurse in a Shakespeare play, the nurse in Shakespeare in Love has no name and nothing to do other than support Viola. She helps her cross-dress, covers her tracks, and provides the audience with humorous comments and pratfalls.
(To a modern viewer, the nurse is even funnier, because she is played by Harry Potter actress Imelda Staunton, who played the fierce and evil Dolores Umbridge. The nurse is as far from Umbridge as you can get, but shows Staunton's versatility as an actress.)