When the pistol is first shown in the dentist's hand, the slide is locked back in the open position. It would be in this position after the last bullet has been fired. In the next shot, the slide is in the closed position, after ejecting one shell and chambering the next, with the hammer cocked (back) ready to fire another round.
Upon inspection of the victim Inspector Japp quotes "revolver grasp in lifeless fingers". The gun is not a revolver but a semi-automatic.
The newsreel at the beginning suggests that the Prince of Wales visited India in 1925. However, the visit actually took place during the winter of 1921-22.
When the body of the dentist is discovered, the corpse opens and then closes his eyes.
As Alastair Blunt visits Mabelle and Gerda in their dressing room (in India in 1924) a stage hand is rigging a Strand Pattern 23 lantern outside the door; these were first manufactured in 1953.
A 1925 visit of the Prince of Wales to India is reported in a movie newsreel with perfect sound attached. Talkies weren't made to this quality at the time. The first talking movie, The Jazz Singer, wasn't produced until 1927, and sound in that was basically only for the songs, and very poor.
In a scene around 1:10:42, as Poirot (David Suchet) and Blunt (Peter Blythe) stroll through the garden, the audience hears Blunt discussing the work that goes into maintaining his hedges. However, the actor's lips do not move throughout the entire monologue, and therefore the line was presumably added in post-production.