In the beginning of the episode we see Prince Phillip and Group Captain Peter Townsend partaking in a brief airplane flight, and it can be clearly seen that the trees below them are fully covered in leaves, suggesting it is a warmer season. However, it is implied that only a few days pass between this event, and The Great Smog of London that lasted from the 5th to 9th of December. In the final scene of the episode we see Peter Townsend and Prince Phillip taking another flight, once again, implied to be only a few days after The Great Smog, and like in the first scene, the trees are fully covered in leaves. This can't happen during a cold month like December.
When Churchill learns that Prince Phillip is learning to fly, he tells Queen Elizabeth that nothing she nor the Royal Highness do is private. Phillip at this point isn't a HRH, rather he is a duke. Its not until 1957 that he was elevated to HRH and granted the title of Prince, five years after the Great Fog.
At the beginning of the episode, Prince Philip notes that he has been made the most senior airman in the country (Marshal of the Royal Air Force). However, later on in the episode, the date is referred to as December 1952, despite Philip not being appointed Marshal of the Royal Air Force until January 1953.
When 'Bobbety' calls Lord Mountbatten, the servant announces him as the Marquess of Salisbury. At the time, 1952, Robert Gascoyne-Cecil was Viscount Cranborne, and would not inherit the title of Marquess of Salisbury for another 30 years, in 1972, on the death of his father.
While the fog at street level would have been impenetrable, underground the air would have been clearer, and people would have taken the Tube safely. (This was documented in an 1898 Parliamentary commission on the quality of the air in the Underground -- and that is back before electric trains, when they were running actual coke-powered trains underground.)
When Kenneth, the chief scientist of the Meteorological Office write to Downing Street, he signs the letter using a Parker Premier fountain pen, a model that did not exist in 1952.