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)
Episode One - S1-E1
Factual error: In the fairground chase the characters jump over some Heras fencing that wasn't invented until the next decade.
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.
Factual error: At both funerals, everyone wears combat dress. Soldiers wear service dress (formal uniform tunic and trousers with collar and tie) to funerals. In fact, combat dress is all anyone ever wears in the series. The British Army has a variety of different uniforms. Combat dress is not worn all the time, especially not by officers or senior NCOs.
You Always Hurt the One You Love - S1-E1
Factual error: In the opening sequence, Dr Frank Winter is hitting golf balls in the desert while a radio announcer is reading off names of Americans killed in action. One name is identified by the rank of "Lance Corporal," which the US Marine Corps first adopted in 1958, long after the war. Several minutes later, Dr Winter dresses down an Army MP, referring to him as "Sergeant First Class." During WW2, that rank was actually "Technical Sergeant."