Plot hole: If the Lady of the Lake walks every night to her room and kills those in her path, why didn't Flora and Miles' parents die in their room by the hands of the Lady in the Lake when they first moved in?
Factual error: The British schoolroom has a UK flag at the front. American schools might do that; UK schools don't. There's even a little one on the teacher's desk.
Factual error: Dominic Wingrave says "it took me 6 years to do the math." Not something any British person would say - in the UK it's always "maths."
Factual error: The phone in the manor kitchen uses a single ring common to North American phones, not the double ring used in Britain.
Factual error: The show doesn't seem sure what Henry and Dominic's titles are. Dominic is referred to as both Lord Wingrave and Mr Wingrave (and his wife as Lady Wingrave). Henry is referred to at various times as Lord Wingrave, Lord Henry Wingrave, Sir Henry Wingrave and Mr Wingrave. Dominic is the elder brother, so if the title was inherited only he would hold it. When he dies, it would pass to his son, Miles, not his younger brother. If they held the titles as younger sons of a duke or marquess (although there is never any indication that they have an elder brother, so this seems unlikely) then they would be Lord Dominic and Lord Henry, not Lord Wingrave. In neither case would they be addressed, especially not by their staff who would definitely know the correct form, as Mr Wingrave or Sir Henry.
The Two Faces, Part One - S1-E3
Factual error: The British policeman carries an American-style long baton on his belt. No British police force used batons before 1994 (the series is set in 1987), and batons of this pattern were never used in any case.
Visible crew/equipment: When Dani hands the spider back to Miles after he uses it to scare Flora, just after she says "Including me" and the spider moves from Dani's hand to Miles' hand, a red tape marker can be seen on the ground right under Victoria Pedretti's left foot. (00:34:09)
Factual error: In the first part of the story-telling the main character appears to be in front of a wrist-watch store's window. That store is Richard Mille, a brand that only started making watches in 1999, the story is set in 1987. (00:06:05)
The Two Faces, Part Two - S1-E7
Continuity mistake: The first time Peter is inside the memory with his mother, two ornamental flags can be seen hanging on a wall next to a light switch as they talk in the living room. One of these flags suddenly turns roughly 45° to the left without anyone touching it. (00:07:33)
Continuity mistake: Hannah's outfits don't match when she follows Miles into the Chappelle after she finds him smoking. She goes in wearing red and once she enters they show her wearing green. (00:24:16)
