Factual error: Mickey and Danny walk to their local shop to buy ice. We see that the shop is Robis, which is at 106 Brick Lane, in E1. They even show the street name sign just so we can be certain. Mickey and Danny deliver the ice to their flat, which is in the street directly opposite the shop - the camera shows them going into the building. However, a few shots later, the whole crew meet on the roof of the building to discuss The Henderson Challenge. We see that this building overlooks the north side of the river, close to Tower Bridge. We get a clear view of the Greater London Authority building which is on the opposite, south bank. They are suddenly several kilometers from Brick Lane, which is where their flat was just shown as being.
Factual error: They plant Albert's hair on a brush used by the Queen Mother in order to have a DNA sample taken from it match that taken from a hair plucked out of his head by Francis Owen, their mark. The Queen Mother's (non-existent) son would not have the same DNA as her. It could be used to establish a biological relationship but it would not be identical - it could not be. First, Albert's DNA is identifiable as that of a male. The lab testing the sample supposed to be the Queen Mother's would see that immediately. Second, a son's DNA is not identical to either of his parents or his siblings (if any) - it is at least 50% different. Any lab worth their fee would realise in a second that the two samples were from the same person. Another problem - Owen plucks the hair from Albert's head and Stacie handles the hair from Albert she plants on the hairbrush with bare fingers, in both cases hopelessly contaminating the samples with their DNA. The tests really are that sensitive.
Factual error: The laboratory technician uses a bog standard light microscope to match the two DNA samples, one from Albert and one allegedly from the Queen Mother. That's absurd. DNA samples are compared using a procedure known as SDS-PAGE, otherwise known as gel electrophoresis. This produces the familiar chart we know as DNA "fingerprints" - bands of light and dark showing the composition of a DNA sample which has been broken up by enzymes. You cannot examine DNA with a light microscope - you couldn't even do it with a scanning electron microscope.
Factual error: In the flashback scene to Whittaker senior's trial at the Old Bailey (which takes place in the late 19th or early 20th century), the judge in the trial is wearing a long, full-bottomed wig. This is completely incorrect - since the late 18th century, full-bottomed wigs have only been worn by judges on ceremonial occasions, not in court. At trials, judges wear short wigs instead. (00:05:17)