Early in the film, Stanley's editor states he has arranged passage to London for Stanley on the steamer Great Eastern. The film at this point was set in 1870. The SS Great Eastern had ceased serving as a passenger ship well before 1870 and was just ending its service as a cable laying vessel.
At the National Geographic Society Stanley states he is English because he was born in St. Asaph. This is actually in Wales.
It is questionable that Livingston would have had the villagers singing "Onward Christian Soldiers". The tune was written by Sullivan in 1871, the year in which the Stanley met Livingston at Ujiji; Livingston had been out of contact with the outside world for several years at that point.
The first part, in Wyoming Territory during Indian wars, was filmed in Idaho, Sun Valley - not Wyoming. You can see the Ski Slope and Ski Runs for Sun Valley Resort in the background.