Christie was arrested by a policeman on the embankment overlooking the River Thames on 31 March 1953, i.e., early spring. However, in this scene the trees nearby are in full foliage (late spring-early autumn) when in reality they would have been bare leaved.
Christie is wearing a police uniform when Muriel Eady comes to the house in the opening scenes but he had already long left the police by this time. Muriel would have known this as she and Christie met when they both worked for the same employer.
The position of the manhole cover changes from shot to shot when the police lift its lid after Evans' "confession" - first it is seen (as it was in real life) in the middle of the road opposite No.10 with the front door to the house closed, but in the next close-up shot we see it being lifted up from a position in front of a bay window from a house further down the street, with the front door of (supposedly) No.10 open
In the Bar scene where Beryl and Tim agree to have the abortion done, Tim is seen drinking from an almost full glass of beer, then when the camera switches, it is almost empty, then it fills back up for the next shot.
At the end of the film where Christie is arrested, the police constable leads him away from Putney Bridge (seen in the background). Putney police station is, in fact in the opposite direction in the Upper Richmond Road. The direction the constable takes Christie in the film would only lead down the river towpath towards Hammersmith.
When Beryl's bound and bundled body is discovered by police, the cord is cut and her legs freely stretch out. That could not have happened in reality since rigor mortis would have long set in. Also, no mention is made of the odor that would have been noticeable.
At the beginning of the film Christie is wearing his medal ribbons for 1914-18. They are back to front, first should be 1914-15 Star, then War Medal and finally Victory medal. The rainbow watered ribbon for the Victory medal is first on his tunic, it should have been last.
When Christie is explaining the procedure he's about to perform on Beryl, he says that natural gas contains carbon monoxide, then quotes its formula as CO2 (which is actually carbon dioxide). The correct formula for carbon monoxide is CO. However, the point appears to be to show him for the half-educated conman he is. It's just Christie's character creating an air of "expertise".
When Christie is murdering Muriel Eady he uses a breathing mask as a gas mask connected to the cooking gas mains, over Muriel's face. But just as she begins to succumb her nose appears above the mask, which would allow her to breathe normally.