Basically, the Royal Standard was NEVER shown at half mast for any reason...not even the death of the monarch.
Until 1997 the only flag to fly from Buckingham Palace was the Royal Standard, the official flag of the reigning British sovereign, and only when the sovereign was in residence at the palace. Even in times of mourning, the Royal Standard would not fly at half mast. The only time a different flag would be flown from the Palace would be upon the death of the sovereign, when the flag of the next most senior member of the Royal Family present at the palace would be raised
However, Princess Diana was incredibly popular with the British public who, for the most part, were unaware of the tradition since most of the population wasn't alive for the last change in monarch.
However, that has now all changed.
This tradition changed in 1997, following the death of Diana, Princess of Wales, when the tabloid press reported alleged public outrage because the palace did not fly a flag at half mast. The Queen was at Balmoral Castle at the time, so there was no flag flying. In response to this display of public opinion, the Queen ordered a break with protocol and the Union Flag was flown at half mast over the Palace on the day of Diana's funeral. Since then, the Union Flag flies from the Palace when the Queen is not in residence, and has flown at half mast upon the deaths of members of the Royal Family, and other times of national mourning such as following the terrorist bombings in London on 7 July 2005, the death of former U.S. president Gerald Ford and the death of Margaret Thatcher on 8 April 2013.
In the actual movie, there was a long discussion about the flag which wasn't even flying over Buckingham Palace and so couldn't be set at half mast.
PRINCE PHILIP Then I hope Robin told him there ISN’T a flag above Buckingham Palace...(angrily jabs coals with a long, sharp fork) ...only the Royal Standard, which flies for one reason only. To denote the presence of the Monarch. Since you’re here, the flag pole is bare. Which is as it should be.
CHARLES clears his throat..
CHARLES Isn’t it possible..that to some people...the Royal Standard is just.. ..a flag? And that the flap pole being bare sends out the wrong signal.
The QUEEN turns. A withering look..
ELIZABETH That’s not the point.
PRINCE PHILIP The point is it’s more than four hundred years old. It has never been lowered for ANYONE..
QUEEN MOTHER Your grandfather didn’t get that flag at half-mast when he died and if your mother were to die tomorrow, she wouldn’t get it either.
CHARLES Yes...but sometimes...in a situation like this..one has to be flexible..