When the transfer of power in Washington is being presented, the British historian narrates: "Dixie, a former musical show tune ironically penned by a northerner, replaces the national anthem." During the Civil War, the USA had no national anthem to "replace." Although "The Star-Spangled Banner" was popular, it was never officially recognized as the national anthem during the 19th century. The Navy adopted it as their song in 1889 and President Woodrow Wilson proposed it as a national anthem in 1916, but it was not given formal status until 1931. In the 1800s, the "Banner" was one of various songs that served as unofficial anthems, including "Hail Columbia" and "My Country, 'Tis of Thee". It would have been more appropriate to have the man in the film say "Dixie, a former musical show tune ironically penned by a northerner, is adopted as the national anthem."
The film seems to be aware that the Confederacy set a six year term for its presidents, by having presidential elections at different points in the story in 1960 and 2002. But there is also one in 1880 which does not fit that pattern.
In the short film "The Hunt for Dishonest Abe", the silent comedian representing Abraham Lincoln has his blackface disguise smeared off his right cheek on one frame before the Confederate general streaks his fingers on that spot.
The basic premise is faulty. At no time during its existence did the Confederate States of America seek to conquer the United States. The CSA's constantly stated goal was to be left alone as an independent nation, i.e. being the big fish in the small pond rather than the small fish in the big Union. Any territorial expansion would have been southward toward the "Golden Circle" of Central America and the Caribbean. It would not even have been remotely feasible to conquer and subjugate the USA, which had much more land and people who were hostile to the South, nor would it have been desirable to do so, as the whole point of Secession was to divorce the South from those lands and people.
In the Summer of 1863, the Confederates capture Gettysburg and Washington, and in the Spring of 1864 General Ulysses S. Grant surrenders to General Robert E. Lee. But Grant was in Mississippi in 1863 subduing the port city of Vicksburg, while George Gordon Meade commanded of the eastern Union army against Lee. In the history happening here it is unlikely that Grant could have made it to the eastern front to supplant Meade.
Before the end credits, the movie explains that numerous products featured as commercials in the fictional world were actual products which were, as the film put it, "Part of the history of the United States of America." However, Darkie Toothpaste, though it did exist, was never an American product, nor was it ever sold in the United States. It was a product manufactured by Hawley & Hazel, a company founded and based in China, and has never been sold outside of Asia. Hawley & Hazel was acquired by the US company Colgate-Palmolive in 1985, which immediately changed the product name to "Darlie" in its English packaging, and changed the face of the black man on the package to a racially-ambiguous one.
Since this film deals with an alternate history which diverged from our timeline in 1863, all so-called anachronisms and factual errors dealing with events from that point on should be regarded as plot holes at best, and redundant at least.
Boston is burned down in the 1860s, and would take a while to regenerate to its previous size. It is therefore unlikely that John F. Kennedy can exist in this timeline. His grandfathers, small children in Boston at the time, would not likely have remained there if they survived the fires, and likely would not have met the mothers of Joseph P. Kennedy and Rose Kennedy.
The existence of Nazi Germany, in a world where America apparently never intervened in the First World War, is highly improbable. Kaiser Wilhelm II's regime likely would have won the war and continued leading a healthy, proud German Empire, leaving no opening for the Nazi party to fill.