When the characters first enter the original Tardis and 12 puts on his sun glasses, there is a wide angle shot showing the actors with the roundels in the background. However, you can see the roundels change shape. The scene is clearly composed from two shots overlaid.
The First Doctor acts as if he is astonished that he will eventually regenerate into another person. Yet, by the time the First Doctor has reached this point in his life, he has already experienced the events of The Five Doctors (1983) and The Three Doctors: Episode One (1972) in which the First Doctor was made very much aware that he would eventually regenerate into several different people. However, it has been previously explained in The Day of the Doctor (2013) and The Doctor Falls (2017) that earlier incarnations cannot retain any memory their meetings with later incarnations.
A black man is shown among the British troops. Though many non-white soldiers did fight, at the time they were always in separate units. During combat troops who have been separated from their units (or who were initially assigned to units which have suffered heavy casualties) will often be co-opted into other units. The soldier in question would, no doubt, be returned to a segregated unit as soon as one became available.
At the beginning of the episode, when the archive footage from "The Tenth Planet" (1966) is playing, it says "709 episodes previously" when in fact there are almost a hundred lost episodes from the classic series, but the episodes that are lost did actually air on television.
When the Captain is taken from the World War One battlefield he has his revolver in his hand. After he falls to the ground at the South Pole, the gun is in his holster and he has to draw it again.
The Captain is clearly shown having his memories copied. We have no idea how long the process takes, and there would likely have been plenty of opportunity for him to holster his revolver.
(Around 30 minutes) The doors of the First Doctor's Tardis open as he approaches them, even though he didn't operate the door control on the console.