The dowager countess is 75 and claims to not know what a gramophone is. This would have been impossible in 1919. The phonograph was invented in 1877 and by 1900 was in every prominent household in the world. The countess was an educated, well traveled woman, she would have seen and heard one scores of times. There was no radio yet, phonographs exactly like the one she saw were everywhere. In movies, in magazine ads, in peoples homes.
When Lord Grantham and Jane say goodbye, her hair is parted on her right for the entire scene except the kiss. In the kiss, is it parted on her left.
Anna and Bates spend their wedding night in Jane's room. Being a housemaid, Jane would have slept in a much plainer room, in a much smaller bed, on a different floor.
The episode takes place in 1919, but the song on the gramophone, Look For the Silver Lining by Marion Harris, wasn't released until 1921.
When the gramophone is first shown (around 8:50) the music plays but the needle is not in the groove.
When Dr Clarkson is discussing Lady Grantham's condition with Lord Grantham, Mrs Crawley is wearing red nail polish while listening to them.