In the conversation with Bertie after delivering the Christmas proclamation, the old King calls Joseph Stalin "Marshal Stalin". Stalin took the title "Marshal of the Soviet Union" only in March 1943, after the Battle of Stalingrad, 7 years after King George V died.
When the newsreel about the 1937 Coronation ends, it is immediately followed by a report of a big outdoor Nazi rally with Adolf Hitler taking the salute, then footage of Hitler addressing an audience indoors. The second Hitler scene is from 1932-3, when he was still trying to appeal for votes. One scene shows SA leader Ernst Röhm, who was disowned and executed in 1934, with all public display of his image banned in Germany.
At the end of the movie, while the King delivers his wartime speech to the nation, a group of men in the next room are listening to it via headphones. The very dusty bakelite loudspeaker cabinet on the table was manufactured in the UK, by the broadcast relay company Rediffusion Limited, in the 1950s.
In a brief scene at the end of the film, a group of soldiers listens to the speech as broadcast. They're wearing the cap badge of The Queen's Regiment, but the badge is the pre-1922 pattern, a paschal lamb with "The Queen's" scrolled beneath. That changed in 1922, to a larger paschal lamb with no scroll.
After the abdication of Edward VIII, Bertie says to Logue "every Monarch in history has always succeeded someone who was dead, or just about to be." That is not entirely true, as there have been some exceptions. The most notable special case is James II, who fled England and was dethroned in absentia in 1688-9, lived in exile in France until his death in 1701, having outlived one of his successors (his daughter Mary II).
After King George VI gives his war speech in the studio with Logue present, he puts his military jacket back on, and the jacket collar is folded down in back as it should be. In the next shot, the collar is flipped up in back as he talks to Logue. In the shot when he leaves the studio, the jacket collar is back down flat as it should be.
When the Duke of York tells his daughters the penguin story, the view from behind shows each girl with an arm around a dog. In the next shot, with the girls shown from the front, their arms are down by their sides.
When Lionel is dining with his family, the face shot of him shows a moderate amount of food on his plate. When it switches to a side shot there is a much smaller amount of food. When it switches back to a face shot, the original moderate amount of food is again shown.
In the nursery, when Bertie tells the princesses the penguin story, Helena Bonham Carter, sitting in the corner, has something rectangular in her midriff under her dress, either a cell phone or a wireless mike transmitter.
Stanley Baldwin is depicted as resigning as Prime Minister on a point of principle, because he has misjudged Adolf Hitler. In fact Baldwin retired from the Premiership of his own free will, the day after King George VI's Coronation in 1937. Hitler had not yet even begun his campaigns of invasion and treaty-breaking at that time.
In a brief scene at the end of the film, a small group of factory workers are listening to the speech on the radio. The speech was broadcast at 6pm on a Sunday, when factories would be likely be closed. If the factory was operating, workers would be unlikely to stop for a short radio broadcast. If they did stop, a much bigger crowd of people would be gathered around the radio.
Towards the end of the credits, it is written: "Piano Concerto No.5 'Emperor' 2nd Movement Composed Ludwig van Beethoven", instead of Composed "by" Ludwig van Beethoven which is the case with the other music pieces in the credits.
When Bertie talks to Winston Churchill the latter asks him what name he would take as king and dismisses Albert funnily. Queen Victoria made a clear statement that in respect of her husband no king will be named Albert. Churchill can't have suggested that he would be came a king Albert.
In the movie, an air raid alarm occurs in London in September 1939. There was a false air raid alarm at 11:28 a.m. on September 3, 1939, the day war was declared on Germany. Genuine air raids occurred the next year.
It's very unlikely that Logue's boys would make plastic models in the mid- to late 1930s. Scale modeling then was done in balsa wood. Bakelite plastic models were available for a short time before WWII, but they were removed from the market for wartime production of other items.
When Logue receives Bertie unattended while playing Shackespeare with his sons, they are sitting in the studio. Logue apologizes them to go to receive Bertie, from the only door to the studio, and welcomes Bertie. Next shot in, the boys are vanished from nowhere.
In the climactic scene where the king delivers his live radio address, the costume is correctly that of the Admiral of the Fleet but it is clearly a costume and not a proper uniform. The fabric is too lightweight and despite the existence of numerous photographs of the king taken that day, the stripes on his sleeves are spaced very incorrectly.