Continuity mistake: After Jody kicks and knocks Bruneau Boyd to the ground, Jesse peels out in the pickup truck and Bruneau's body disappears, even though it should be lying next to the truck as it quickly departs.
Factual error: The show starts in 1901 with a woman constable working for the Metropolitan Police Service (Scotland Yard). However women didn't join the Metropolitan Police until 1919 (which is also when the first woman's police uniform was created). Prior to that, women volunteered in the Women's Police Service, but that wasn't formed until 1914.
Factual error: Ray, Sara, and Kendra are Left Behind in 1958. However, 10 weeks later Sara and Kendra are playing the board game "The Game of Life." The modern version, as seen, didn't come out until 1960 (previously it was called The Checkered Game of Life and doesn't resemble the modern version).
Factual error: With two exceptions, all the policemen shown in both 1949 and 1953 are wearing the old-fashioned tunics with high, closed collars. The Metropolitan Police adopted the more modern open-necked tunic with collar and tie in 1948, and this is shown worn by the policeman who visits Christie and by the officer who arrests him.
Factual error: Bozer and for that matter I would assume all of the team went to a spycraft academy. A basic subject in any of such academies is surveillance, including proper techniques to follow and stake out someone. Bozer being so inept at tracking MacGyver's father is inconceivable given their training.