The story takes place over a four year period. However, Greville's son Andrew doesn't grow or change at all over a period (ages c10-14) he would be entering puberty.
The movie takes great pains to paint Greville as merely a courier who didn't look at the documents he was delivering. Yet, when his superiors tell him he's done, he asks about the situation in Cuba, which he couldn't have known about. This is before the situation became public.
When Greville is in the airplane trying to leave The Soviet Union for the last time, from the window he sees a dark car stopping beside the plane. The police, KGB or GRU has obviously arrived. The car is easily recognizable as an English produced Bedford model CA (1952-1969). It is not credible that Soviet authorities are using this type of vehicle.
When Wynne is visiting Moscow, the name of the hotel is "Vitaliy", which is a Russian male name. There was no such hotel in Moscow in 1960's. The naming convention at the time would not allow for such a hotel to exist. Moreover, a foreigner visiting Moscow would stay in just a number of hotels, such as "Rossia", "Metropol" and "Ukraine" -- all of them were under an umbrella of a government company "Intourist" (which is basically short for "foreign tourist" in Russian).
At the end of the film, it states that in 1963 a telephone hotline was established between Washington and Moscow. This is a common misconception. The hotline was in fact a teleprinter, there never was a telephone hotline.
During his first meeting with the Western agents, Col. Penkovsky pulls a Soviet document out of a briefcase he didn't bring to the meeting.
The Sound of Music marquee shown in the film is of the stage production, which started its run in London in 1961.
Wynne is seen boarding a BEA Vickers Viscount for his trips to and from Moscow. The Viscount (which is never seen flying) did not have the range to make the trip; in 1961-62 the regular BEA service to Moscow was flown using a De Havilland Comet 4. IWM Duxford has a Viscount in BEA livery but there are no surviving Comet 4s in BEA livery.
Wynne and Penkovsky go to the ballet on two different occasions, but each time the same woman, dressed the same way, is sitting behind them.
The shot of the soup Wynne is eating in the prison is used twice. The pieces of the vegetables are the same in two different meals.
When Greville is taken out of the plane in Moscow Airport, he is thrown into a van. This van is a Bedford model CA, made in the UK, and is not likely to be in Russia at that time. It should have been a UAZ, a most common Russian van.
In a panoramic shot of Moscow, Wynne walks along the embankment opposite the building of the Ministry of Defense. Although it was built in 1951, a late 2014 annex with two-headed eagles on the friezes is visible. In Soviet times, two-headed eagles were a symbol of czarism and under no circumstances could they appear on an official building in the center of Moscow.
The movie portrays Penkovsky witnessing the execution of colonel Popov. While this is debatable, Popov was executed in January 1960. The first scenes of the film take place in August.
At least twice in the movie, someone speaks about "gulag" as the worst outcome of Greville's mission. The word "gulag" got an international repute with the publication of Aleksandr Solzhenitsyn's The Gulag Archipelago in 1973. Or, the story of The Courier in situated in 1962.
The black cabs seen in the London street scenes hail from later than the early 1960s when the movie is supposed to be taking place. They have the larger tail lights and are missing the bunny ears high mounted light units sported by these cabs from this period.