In the episode it is suggested that Earl Mountbatten had not seen his sister, and the Queen advises him that his best service to the Crown would be to visit Princess Alice. In reality, historical records show that at that point Princess Alice had been a frequent visitor to Mountbatten at his home - Broadlands - since the end of World War II.
The episode makes it appear that there was tension between Earl Mountbatten and the Labour Government. In reality, Mountbatten was a strong supporter of Labour and he was frequently derided in Court circles as "pink" - that is being soft or sympathetic towards Communism. About the only thing correct in the portrayal of Mountbatten is that they used the correct name for his secretary - John Barrett.
Mountbatten is shown being fired by the prime minister, Harold Wilson. This gives Mountbatten motive to resent Wilson and want revenge. In reality, he was fired by Denis Healey, then Secretary of Defence.
At the General Salute, Mountbatten and his staff give the US salute with the palm parallel to the ground. The Royal Navy salute has the hand inclined forward so that the palm is hidden and the back of the hand is slightly visible from the front. The British Army, Royal Air Force and Royal Marines all show the palm to the front.
Though the year is specifically mentioned as being 1967, on three occasions the cars depicted are from later times - at 28:20 minutes in, the blue Lincoln Continental that Queen Elizabeth II is riding in is a Mk5 from about 1977; at 32:40 the fourth car in the convey is a black Jaguar XJ6 Series 1 released in 1968 (or the virtually identical Daimler Sovereign Series 1 from 1969); and, finally, at 22:03, while the two black Citroen DS Series III 'swivel-headlight' examples could technically be included since they were launched in late 1967, it's more likely that they would have been the earlier Series II single-headlight variety.
When Mountbatten recites the Kipling poem, he says, "Where there are no Ten Commandments." The line is actually "Where there ain't no Ten Commandments."