After FDR met with the Boy Scouts for a photo shoot, Louie shows Eleanor a newspaper article about the event; but neither of the photos in the newspaper corresponds to the poses that were portrayed.
When the attendants lift FDR into the pool, the reflection in
the water shows Kenneth Branagh's two completely healthy legs, instead of the computer-generated images of legs used throughout the film to represent Franklin D. Roosevelt's disfigured appearance.
You can see when the train is pulling into the station the words "Tennessee Valley" which is the Tennessee Valley Railroad in nearby Chattanooga, Tennessee.
In the opening scene, newsreel from the Democrat National Convention of 1920 shows FDR talking and people cheering as part of the film. Film with sound wasn't invented until 1927, and even then was quite rare until 1930 or 1931.
At the 1928 Convention, several people are shown waving 50-star flags, not introduced until 1960. From 1912 through 1959 the flag had 48 stars.
In Roosevelt's Warm Springs office, during the sequence where Louis Howe and then Eleanor Roosevelt attempt to convince FDR to nominate Al Smith at the 1928 convention, sitting on a book shelf is Winston Churchill's six volume account of World War II, which was not published in its entirety until 1953; 8 years after Franklin D. Roosevelt died and 17 years after Louis Howe died. (The spines are pink/red and distinctive.)
Hospital patients in the 1920s sing "I Won't Dance," from the London musical comedy Three Sisters (1934).
In 1921 before his infection, FDR is seen dancing with his private secretary to a 1939 record: Alice Faye singing "I'll See You In My Dreams" from the soundtrack of La rosa di Washington (1939).
The train seen in an external shot as the Roosevelts travel to Georgia for the first time is clearly not of the same type as the train they are in. The train seen externally is British not American.
At the start of the film, when FDR is addressing the delegates to the June-July 1920 Democratic Convention, he states "Just ask my cousin Teddy!" regarding his own political chances as a Roosevelt, an obvious reference to his distant cousin Theodore Roosevelt. This remark is a bit ghoulish, as Teddy had died in January 1919, just over a year earlier.