At 1.06 min, Holmes takes the revolver. A cocking click is heard, but the pistol is clearly not cocked.
We are shown two newspaper front pages. The date of the first newspaper is Tuesday October 2, 1886. The actual day was Saturday 2nd. The second newspaper was dated Wednesday October 17, 1886. The actual day was Sunday 17th.
When Holmes and the police search the home of a suspected murderer at the beginning of the show, they are using battery-powered flashlights in 1886.
They are in fact using carbide lamps...similar to what miners used in the same period.
Holmes is shown reading "The Guardian" newspaper. In 1886 it would have been called "The Manchester Guardian". It didn't become "The Guardian" until 1959.
Holmes describes the killer targeting opium dealers as a 'serial killer'. this term was not coined in English until 1966. Even its earliest form--the German Serienmörder ('serial-murderer')--was not used before 1930.
Moriarty would not be able to fall from Big Ben directly into the Thames as it is some 50 meters from the east clock face.