Alan despite arriving at the opera in a dress, he is seen running through the opera house after the battle in a suit, even though in the timeline given he would not have had the time to change.
A group of characters work in a modern day office setting despite the story being set in the Victorian Era. This is possibly explained by the fact that they exist in a universe parallel to the living world, which is a timeless purgatory of sorts.
The characters are seen multiple times at Phantomhive estate, despite stating multiple times that they are in London. The estate is in the country outside London. They should have been situated at the townhouse in London instead.