Whilst in the South Pacific, Harry has no trouble making a phone call to Britain. In 1963, before satellite communications, this would in fact have been very difficult and time consuming. He would have had to call the operator and book a call, which would have had to travel over copper cables. At that time you could not even call the USA from Britain without booking a call with the operator.
B52 squadrons are shown as being based in Europe. All SAC B52s were based in the USA. There were never any B52s permanently based in Europe.
The CIA offer a British agent a salary of $1500 pa to join them. In 1963 £1 was worth £2.80, so a salary of $1500 was worth about £535 pa. To put this into context, in 1964 a London bus driver's salary was £904 pa. The salary the CIA was offering was insultingly low, and therefore is hardly plausible.
When Harry is shown dossiers of agents in Lebanon, they are listed as speaking various languages, such as Arabic and English. Yet none are listed as speaking French, which was and still is an important language in Lebanon, as it had been a French territory. Also, the dossiers state there is no photo for some of the agents, when the photo is attached to the dossier.
The Russian city of Nizhny Novgorod is mentioned by one of the agents. It is mispronounced as "Nizhgy Novgorod". Furthermore, between 1932 and 1990 the city was renamed by the Soviets as Gorky, so the name Nizhny Novgorod would not have been used in 1963.
A government minister is seen signing a document with what is clearly a modern ballpoint pen. In 1963 a fountain pen would have been used for official documents such as this.
Whilst Harry is at the US test site in the South Pacific, we see three Iowa class battleships sailing nearby. In 1963 all 4 Iowa class battleships, and indeed all US battleships were laid up in the reserve fleet. One, USS New Jersey, was brought back into service in 1967 during the Vietnam War, and all four were reactivated in the 1980s, but there were none sailing in 1963.
In the series a twin engine aircraft has been used, once when the Professor is flown away, and once in the final scenes: they used the same aircraft and it is a Cessna 340 that was in production between 1972 to 1984.
Harry's eyeglass frames are modern "retro" style and the lenses have anti-reflective coating not available until the 1980's.