Episode #1.5 - S1-E5
Factual error: Catherine is told she's been recommended for the Queen's Police Medal for Gallantry. This has not been awarded since 1977, when it was replaced by the Queen's Gallantry Medal, which is still awarded.
Factual error: Constantine's driving licence is shown - it's a UK licence, but no. 7 where his signature should be is blank, and no. 8 where his full address should be simply says "England, United Kingdom." Both are printed onto the licence when it's issued - without that info it would never be issued in the first place. And it's not a fake, because why would he have a fake licence with his real information on it? It's just an incomplete prop.
Factual error: Congressman Tate and his brother are on the shooting range. They are wearing high-tech electronic earmuffs that block harmful noise like gunfire while amplifying speech and other low-volume sounds. Although, they are wearing high-tech earmuffs, they both have an ear exposed so they can hear each other talk. (00:13:00)
Factual error: 1947-1948 Ford Sedans and 'woody' station wagons are regularly shown, rather than earlier models - even 1940-1941 Ford vehicles. A simple Google image search will show you the difference. (The atomic bombs were dropped on Japan in 1945).
Factual error: A number of times throughout the season they refer to LSD being found in dead bodies and LSD being matched with other LSD found in other (living or dead) bodies. LSD starts a chain reaction in the brain which requires about an hour to take a noticeable effect. And, by that time, it has completely broken down and no longer exists as LSD in the body - before the person even begins to get high. There is no test for LSD even in the most bombed living individual, let alone a dead body years later.
Factual error: Helmuth von Moltke is addressed and referred to several times as Field Marshal. He wasn't promoted to that rank until 1871. He was a lieutenant-general during the Second Schleswig War in 1864.