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.
The goof items below may give away important plot points.
When Mr Bates is arrested by police officers the handcuffs used are a ratchet type, which were not in use by British police until the 1950s. The correct type should have been a Hiatt Darby or a Dowler Darby non-adjustable single locking key lock model.
During Lavinia's funeral, Thomas is seen wearing his footman's livery, complete with white tie, underneath his overcoat. Like the rest of the men present, Thomas would have been wearing his usual church-going suit, complete with a black cravat as befits a funeral. There's no way he'd appear at a funeral wearing white tie.