When Churchill is summoned by the king, as his car drives onto the the grounds of Buckingham Palace, the Union Flag can be seen flying above the Palace. At that time the only flag to be flown over the Palace was the Royal Standard, and only when the monarch was present. This rule was not changed until 1997, and even today, the Royal Standard is flown whenever the monarch is present.
Four RAF fighter planes are seen flying over London in a finger four formation. While Luftwaffe had started using that tactic earlier, the RAF used a three plane "vic" formation until the end phases of Battle of Britain.
Live broadcasting from Parliament did not begin until the 1970s so his speeches could not have been heard on the radio as they were being made. He repeated some of them later for the BBC and these are the recordings that are now available.
At some point of the movie, you can see a Quartz clock (It being Quartz can be understood by the movement of the seconds hand. In mechanical clocks the movement of the seconds hand is smooth with step sizes as small as 1/6th or 1/8th of a second, while in Quartz clocks usually the seconds hand jumps for a step size of 1 second, and then pauses until it makes its next 1-second step jump for the next second).
The Quartz clocks were of course invented before the time of the movie. But as late as 1960s that the development of cheap semiconductor digital logic made their public and domestic use possible, their use was just limited to some specific applications (like some scientific laboratories and so on).
While the C-47 Skytrain had it's first flight in 1941, the Douglas Corporation DC-3, upon which the C-47/Dakota was based on first flew 17 December, 1935. The DC-3 was one of, if if not the, most successful twin-engine, propeller-driven airliner of its day. When War 2 began, the British government impressed hundreds of civilian aircraft into service for the duration of the war. Thus, it is probable that a DC-3 in RAF livery would have been extent in the Spring of 1940.
In the House of Commons when Churchill first appears as Prime Minister, we learn that Chamberlain is to wave with his handkerchief if he wants to signal support for Churchill's policies. Just before Churchill stands to make his speech, Chamberlain can be seen placing his handkerchief on his knee. As Churchill speaks and the camera pans around, Chamberlain's handkerchief is back in his breast pocket. When the speech has concluded, Chamberlain's handkerchief has reappeared on his knee.
When on the Underground, Churchill is advised to travel to Westminster on the District Line. This is a sub-surface line that uses carriages of normal size. However, the train he gets on is of deep-level 'tube' rolling stock which are not used on the District.
Blackout restrictions were imposed starting in September 1939 and strictly enforced, requiring all vehicles to be fitted with slotted covers that only allowed a tiny sliver of light to be directed downwards toward the road. However, all the vehicles in the street scenes had fully exposed headlights.
Churchill says Edward Wood, 3rd Viscount Halifax would never have turned down the offer to be prime minister as he was the fourth son of an earl. However, Halifax was the son of a viscount, not an earl, a mistake that the real Winston Churchill would not have made. Edward Wood was made 1st Earl of Halifax in 1944. Though he was indeed the youngest of four sons, all his brothers died young. From the age of 8 years old, he was his father's sole heir, and thus he would not have grown up feeling deprived of a dignity by order of birth, as is implied in the dialogue.
The Labour Party did not recommend Winston Churchill to replace Neville Chamberlain. And they do not do so in the movie either; it is merely stated (by Conservative politicians) that he is the only one the Opposition will tolerate to lead a coalition government.
The conversation between Churchill and the public while riding the Underground continues for more than 5 minutes, but the journey from St James' Park to Westminster takes less than 2 minutes. However, during a war, there are many reasons why the train could run slower.