Factual error: The first Kinetoscope (arcade of 'personal movie viewers') was opened in 1894 and the first motion picture studio was located in New Jersey. There is no way that they would have known enough about motion pictures in 1887 to go into them, let alone go to Hollywood to do so. And the use of the word 'movie' is at least twenty years too early.
Factual error: Several real-life events are off in the movie's timeline. The movie itself is set in 1887, yet it includes Arthur Conan Doyle beginning writing Sherlock Holmes, although he did so before 1887; a mention of the Savoy Hotel, which was not built until 1889; and many references to and an appearance from Jack the Ripper, who was not active until 1888.
Factual error: Near the start, we see a long shot of New York Harbor, where we can see the Statue of Liberty under construction. When we see the statue it's green, which comes from the statue corroding, which it hasn't had time to do.
Factual error: In the scene where Owen Wilson and Jackie Chan confront the street urchin who stole Wilson's watch, Wilson tells him they have "parents who love us", but as he says it he is looking over way over the boy's head. [If you watch the outtakes, Owen Wilson says he feels "so mean" talking to a child that way, so it's possible he simply couldn't be that mean to a child while looking him in the eye. Or he could have been trying to keep from laughing. Either way, it's still a valid mistake, this is just a possible explanation.]
Factual error: During the scene when they are about to try to drown Jackie & Owen you can see the ceiling. On it are large shop lights. Even though lights were invented over 50 years beforehand, lights that new looking and of that kind were not.
Factual error: In the fight scene where Jackie Chan and Donnie Yen fight on the boat right after Yen starts firing the gatling gun, Chan kicks him away from the gun and the gun keeps going. A gatling gun stops firing when the trigger is released, and a new gun such as that one isn't going to have problems with it "sticking" down.
Factual error: Before he became the famous writer, Arthur Conan Doyle was an ophthalmologist, not a detective.
Factual error: In one of the first shots of London, there is an aerial view of the River Thames that shows the bridge leading to Big Ben and father upstream Tower Bridge. 2 problems: the movie is set in 1887 and Tower Bridge was not finished until the year 1894, and secondly, from that shot, Tower Bridge would be below Big Ben, off screen, not above it.
Factual error: It is British law that a non-British (or Commonwealth nowadays) citizen does not receive the title of "sir" as part of their knighthood. Therefore, Chon and Roy would not be referred to in this manner during the knighthood ceremony.
Factual error: All the cars used in the film have pneumatic tyres. The film is set in 1887 and pneumatic tyres were not successfully used on cars until 1895.
Factual error: Early, rotating-barrel machine guns were all activated by cranks, not buttons or triggers as shown in the film. While there were prototypes of trigger-type machine guns such as the Maxim, under development at the time, they were very different in operation than the Gatling-type gun used in this movie.