Factual error: The scene including the appearance of Joseph Merrick is not correct. The scene shows us Frederick Treves presenting John Merrick to the other doctors. He calls Merrick 'Joseph', but someone corrects him that he should call him 'John'. That is not correct: in real life Treves was the only one who instantly called him John, while his real name is Joseph.
Factual error: The first time you see the Royal London Hospital, the sign says "The Royal London Hospital". In fact, the hospital has only been known by this name since 1990. Before that, and certainly in 1888, it was called The London Hospital. Many people in the East End of London still refer to the hospital as "The London".