During the time travel sequence, the voice-over from King Edward VIII's (Duke of Windsor) abdication speech precedes Franklin D. Roosevelt's "Fear itself" speech. FDR's speech came first (4 March 1933), then Edward both ascended and abdicated in 1936.
Although the Whitechapel Murders occurred in the Autumn of 1888, while the London arc of Time After Time is set in 1893, the script specifically states "the Ripper has returned," so this isn't really an error.
Someone living in late-Victorian England would have been used to guns being readily purchased legally, much as in 1979 San Francisco. It was only in 1920 that gun laws in Britain began to be significantly tightened.
During the scene where H.G. Wells is in McDonald's and sits down to eat the food he comments: "Pomme frites...fries are pomme frites." However, it is clear Malcolm McDowell says the British term "chips" which is later dubbed to "fries" in post production.
H.G. Wells travels from 1893, when he is explaining his machine he says, "12 years ago a French Engineer harnessed the power of the sun to run a printing press." This would have been 1881, but in fact Monsieur Abel Pifre demonstrated his solar powered printing press on August 6th 1882.
The jeweler states it will take a week to get a check and Wells states he doesn't have a week. Wells has a time machine. It would not be hard for him to go to November 14 or 15, collect the check, cash it, and return to November 7.
Before John Leslie Stevenson (Jack the Ripper) asks H.G. Wells for the return key for the time machine and attacks him, he locks the door to the Hotel room. A maid knocks on the door and then opens it saying, "Room service." The door should have still been locked.
When H.G. Wells travels through time, the windshield of the time machine cracks. But when the time machine is in the museum the windshield is not cracked. - Not windscreen but glasses which are broken. He replaces them on arrival at museum.
Though it makes good drama, it is unnecessary for H.G. Wells to chase Jack the Ripper into the future at all, since he could simply travel back a day or two and prevent Jack from leaving in the first place. - There is no evidence in the film that changing history is that easy or indeed possible. Everything that transpires happens in a linear framework. HG travels in time but at no time changes it. Any attempt to do so might have proved paradoxical.
When Jack/John takes off H.G. Wells's Glasses before hitting him, the camera focus is thrown to show how poor H.G.'s vision is. Although Jack/John's face is badly out of focus the wall and door behind him comes into sharp focus, displaying a possible error by the focus puller - to make this effect obvious; everything should be out of focus. However; this effect could be seen as showing Wells as being long-sighted.