Before going out to the balcony, the king smokes part of a cigarette. He then straightens and pulls down his tie. This action exposes his uppermost shirt button. Camera looks at Queen Mary then back to the king. His tie is correctly tied.
Elizabeth asks Philip to stop smoking, as she hated it. The real Elizabeth was not averse to smoking. She used to smoke herself, but only in private.
King George asks Elizabeth to take on a tour in his place for the first time. But by now, the real Princess Elizabeth had already done a tour of Canada with the Duke, filling in for the King and Queen in 1949/50.
Charles and Anne are shown in Malta while Philip was stationed there. In reality, the children were at home in the UK with their nanny.
When Princess Elizabeth leaves Clarence House for lunch at the Palace, the car she hops into is displaying the Royal Standard as well as the St George statuette on the bonnet. Only the reigning monarch would display these.
HM The King refers to himself as the "King of England" whilst waking Philip up to go shooting. The Kingdom of England ceased to exist in 1707 when England and Scotland joined together in union to form the Kingdom of Great Britain (Ireland later joined in 1801 and finally in 1922 when the Republic of Ireland was formed and broke away from the union, in 1922 the King became the King of the United Kingdom of Great Britain and Northern Ireland) . It would have been more appropriate for the King to refer to himself as "The King" or perhaps as "The King of Great Britain" (although as stated above that too would be incorrect).
Philip complains to Elizabeth that the world tour in 1952 will take them away from the children for months. In reality, his duties in Malta, with his wife accompanying him, had already taken them away from the children for many months.
In the hunting scene, the shotguns noticeably do not recoil against the king's shoulder.
At around 40:46 (and especially at 40:48) you begin to see a light blue outline around Prince Philip's left side of his coat facing the window, revealing that this is being shot in front of a blue screen.
If the King were spitting blood, it is very unlikely that he would dismiss it so lightly. Tuberculousis was a very real threat still, and any symptoms would have been checked quickly.
Prince Philip's mother, Princess Alice of Greece, did not found her order of nuns until 1949, two years after the wedding of her son. She did not wear a nun's habit to the wedding and can be seen clearly in the photographs of the wedding party wearing a full-length dress. However, she wore her habit to the Coronation in 1953.
When Prince Phillip was in naval uniform for the wedding he was wearing naval flying wings on his left sleeve above his rank. However he didn't learn to fly until later. The official wedding photographs of the real wedding do not show the wings.