In season 1 Polly's daughter is referred to as Sally. In Season 2 episode 2 she is referred to as Anna.
Previously Polly said that her son was five and her daughter was three when they were taken from her, and she said that her daughter would now be 18, but when Polly wants to know where her son is, Tommy tells her she'll have to wait for him to turn 18. He should be at or near 21 if his younger sister would be 18.
When Thomas tells Arthur to look at himself, Arthur throws the vases in front of the mirror on the ground. In the next shot, when Thomas leaves the room, the vases are standing in front of the mirror again. When Thomas re-enters the room, the vases can be seen in the background standing closer to each other.
When Polly gets the revolver out of the box and points it at Tommy, she clearly cocks it by pulling back the hammer. When she comes around the desk, the gun is not cocked.
In series 1 Polly tells Ada about how her daughter Sally and son Michael were taken from her by the authorities, but in series 2 when Tommy tracks down their files they are called Anna and Michael.
Alfie Solomons talks of how he killed an Italian during the war. The Italians fought on the same side as the British (who presumably Alfie fought for) in World War I.
Churchill reads aloud a letter from Tommy, stating he fought at Verdun; Churchill also asks who else in the room fought there, to which a few men raise their hands. In actual fact, British forces were not at Verdun and opened up the Somme largely to relieve French forces. Verdun was solely a French effort.
Winston Churchill and several of his assistants claim that that they served at Verdun. There were no British troops at Verdun. It was a battle between the French and German armies. In fact The Somme was a British offensive with the specific objective of taking the pressure off French troops at Verdun.
It is said that Aunt Polly's son Michael has to wait until 18 before he comes of age and can leave his adoptive mother. However the age of majority in the UK was 21 until 1970.
Pol smokes a filtered cigarette. The first cork-tipped cigarette didn't come out until 1936.