Before the wagon train leaves Independence, they all sing "Onward, Christian Soldiers"--which Sir Arthur Sullivan didn't compose until 1871.
The film is set in 1844, yet throughout the battle at the river, several scenes show the use of Springfield "Trapdoor" rifles. This firearm was not introduced until 1873. In addition, what appear to be Colt SAA (Single Action Army) cartridge revolvers are used. This revolver was designed in 1872, and issued to the US Army in 1873.
Gail Patrick bakes Billy Lee a birthday cake, but there's no indication of how she did it on the open plains without an oven. Actually, there are methods of baking cake batter over a campfire, including inside a Dutch oven. However, it is less likely that such a cake would be square-sided as shown.
Although he refers to them as trappers, Jim mentions the cannibalism of the Donner "Party"--which didn't occur until 1847.