We're told thieves broke into the church on Thursday night, but in the flashback, the church is lit by daylight.
When the aliens are morphing, a bright pink light surrounds them. Yet the light is not present at the dinner when one of the guests morphs.
Donna goes upstairs to the locked room with a large magnifying glass. Later, as she walks down the corridor, it is not in her left hand and she uses her right to open the door (which would be difficult if it was in that hand). When we next cut to her, she has that most useful glass again.
The story does not accurately depict what happened to Agatha Christie in 1926. However, The Doctor tells Donna that time is "in flux" due to the alien presence. This excuse is often used to explain minor alterations to history within the Whoniverse.
Agatha Christie went missing in December 1926. However, from the weather and the clothing the guests are wearing, it is clearly not wintertime in the UK.
When Donna enters the locked bedroom that has been locked for 40 years, she picks up a teddy bear in 1926, The 1st teddy bear was made in 1902 so that teddy bear could not been there if the room been unopened since 1886.
The Doctor said there are no policewomen in 1926, but the first British police woman was sworn in in 1915.
Agatha doesn't know what Donna means when she tells her that her books may be turned into films. Films were very common by 1926. Agatha Christie's work was first filmed as The Passing of Mr. Quin (1928) in 1928. Agatha might have been confused by Donna's reference to talking pictures, but sound films were right around the corner. "Talkies" began to be produced sparingly in 1927 (after at least one attempt in 1905 had failed), and had completely phased out the silent films by 1932. It is unclear why Agatha, who had a healthy imagination for creative speculation, should have had so much difficulty grasping this concept.