Factual error: The action takes place around the second half of June (either side of the Glastonbury festival weekend). At the beginning of the film leaving work early evening in time to get home for dinner, in the UK, it wouldn't be dark. Later on, there isn't that much snow in the Alps, especially below the tree line, at that time of year.
Other mistake: At the beginning, as Eggsy is pursued through city streets by the Golden Circle henchmen, the three pursuit vehicles are all firing rooftop mini-guns at Eggsy, in spite of the fact that they are all in single-file, one vehicle behind another. The Golden Circle vehicles would have been riddled with their own gunfire.
Continuity mistake: When Eggsy is fighting in the car and the robot hand grabs his pistol, from one angle the slide is back (almost like the magazine is empty) but from another angle the slide is forward and ready to fire.
Other mistake: One of the robot dogs spits a shell casing after being shot in the mouth. Casings are not propelled but discarded. It definitely made the sound of a casing and not a round so it may just be an audio error.
Continuity mistake: Bennie doesn't attack when Jet attacks because he is charging. He leaves his doghouse when he is full, but seconds later, he is already down over 30%.