In a brief scene at the end of the film, a group of soldiers listen 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.
In the conversation with Bertie after delivering the Christmas proclamation, King George V calls Joseph Stalin "Marshal Stalin". Stalin was awarded the title "Marshal of the Soviet Union" in March 1943, after the Battle of Stalingrad, more than 7 years after George V died.
When the newsreel about the 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 at least four years earlier, when he was still trying to appeal for votes. One scene shows SA leader Ernst Röhm, who was executed in an internal purge, with all public display of his image banned, nearly two years before the actions in the movie took place.
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.
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.
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. For example, James II had to flee England (a de facto abdication) in 1688. He lived in exile in France until his death in 1701.
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.
Each time Queen Elizabeth enters the elevator, she pushes the "down" button to go to the speech therapist's office. The elevator is clearly going down each time. However, his office has huge sky lights and windows, indicating it is on an upper floor.
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.
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.
Stanley Baldwin is depicted as resigning as Prime Minister on a point of principle, because he has misjudged Hitler. In fact Baldwin retired from the Premiership of his own free will, the day after George VI's Coronation in 1937. Hitler had not yet even begun his campaigns of invasion and treaty-breaking at that time.
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.