Paul goes back to 1881 wearing a modern suit, and when he gets there he is dressed in the clothes of the day.
When Paul Driscoll and Prof. Eliot are standing and talking in front of the time machine, there is a device that looks like a tape recorder operating. The amount of tape left on the right reel changes from very little remaining to a lot remaining, back and forth a few times.
A single B-29 was not sent to bomb Hiroshima. Many others were sent as decoys.
The U-boat that launches the torpedo at the RMS Lusitania has a spinning sonar mast, unlikely for a submarine in 1915. Obviously it is stock footage of more contemporary vintage to 1963.
The Japanese police officer Driscoll meets in 1945 wears the long, TV-screen shaped glasses that were fashionable no earlier than the late 1950's.
When the horn player is sitting on the steps of the gazebo, in the background, there appears to be a modern multi-story office building.
When the submarine fires the torpedo, the wire that supports it and guides its path can clearly be seen protruding from the front of the submarine model.
Driscoll wants to assassinate Hitler, yet the time he picks to do so is when Hitler is at the height of his power and fully protected by soldiers and the Gestapo. It would have been much easier to go back to a time when Hitler was not yet a public figure, let alone dictator of Germany.
When speaking with the captain of The Lusitania, Paul refers to The Old Head of Kinsale as an "island." This is incorrect, it is in fact attached to Ireland and is properly referred to as a "headland" (a narrow piece of land that projects from a coastline into the sea).
When Paul Driscoll returns from his failed attempt to alter history, his colleague Professor Eliot states that the past cannot be changed. A few moments later, when Driscoll states his intent to go back to 1881 to live, Eliot cautions him that he could drastically alter history if he changes one thing - the complete opposite position to his earlier stance.
When Paul Driscoll is ready to activate the time machine, he tells Professor to push the button. The professor turns a dial to activate the machine. Driscoll would have known it was a dial, not a button.