While speaking to Mrs. Hughes about London, Mr. Bates says "I haven't been here since I buried my mother...." and he turned from this trip in series 2 episode 1. Later that episode when he leaves with Vera, they return to London to live in his mother's house. Later in season 2, when he visits London to speak to Vera (and she commits suicide to frame him for her murder) she is still living in his mother's house. This is further confirmed in series 3 episode 1 when Anna and Mrs. Hughes travel to London to clean out the house.
When Harold is talking with Madeleine in the park, she alternates between sitting straight up and leaning back on her straightened arms between shots.
At the picnic at the Albert Memorial, the statue of Prince Albert is gold. In 1923 it was black. It was only restored to its present gold appearance in the 1990s.
Violet uses the term "whodunit" in 1923, but that word is not documented before 1935.
It's revealed Michael Gregson may have been killed during the Munich Beer Hall Putsch - the Nazis failed attempt to seize power. However, this event took place in November 1923. That's still about four months away. Still, he could have been killed by Nazi stormtroopers in an altercation earlier. Violence by political factions continued since 1919.
In the ball scene towards the end of the episode, we clearly see that only a string trio is playing, yet the soundtrack is obviously that of a lush, full string orchestra.
The Levinsons and Lord Aysgarth are seen picnicking in a grassy area in front of the Albert Memorial in London. In reality that area isn't grassy at all, but rather a series of concrete steps leading up from South Carriage Drive to the memorial.