At around 1 hour 10 minutes, Sherlock crosses Vauxhall bridge with a double decker bus approaching him. The camera angle changes but there is no sign of the bus which should have just passed him.
When John alights the bus, he uses the door at the front of the bus. In London, passengers board the bus using the front door and alight using the rear door.
When Sherlock first sees the Thatcher shrine, it contains a framed invitation to Mr. Welsborough from the Prime Minister in 1987. At the top, it has the arms of the United Kingdom and then in large letters says, "The Prime Minister The Rt. Hon. Mrs. Margaret Thatcher, F.R.S., M.P." followed by the invitation to a reception. However, this is an incorrect form of address, as the "Right Honourable" before a name replaces Mr., Miss, Ms. and Mrs. and is never used with it. Additionally, letterhead from the Thatcher era either simply had "10 Downing Street" below the coat of arms, or, for a personal invitation to an event, always stated both "The Prime Minister and Mr. David Thatcher" followed by the usual "request the presence of..." and the name of the invitee.
About 5 min into the episode, Dr Watson is seen typing his blog. In the shot, it is clearly visible that the blog we see on the screen, the one he's typing, is actually an image of it opened in Windows Photos app.
Mary's American passport, shown under the name "Gabrielle Ashdown," contains so many errors that it is essentially useless. First, at the top row for passport type/code/passport number, it shows "P USA US3053333" but American passports do not display "US" before the passport number. That the passport number is only seven digits instead of the usual nine would also be a red flag. Secondly, it shows dates using abbreviations for the month in English and French, and a two-digit year, neither of which is correct. For example, her passport shows a date of birth of "16 APR /AVR 75" while a real passport would show 16 April 1975. (This generation of American passports also uses lowercase type for months.) Thirdly, it has an issue date of "11 OCT /OCT 14" with an expiration date of "11 OCT /OCT 24." American passports are valid for 10 years, not 10 years and one day, so the accurate expiration for a passport with an issue date of 11 October 2014 should be 10 October 2024. Fourthly, all the French terms in fine print that should have accent marks are missing them (Nationalité, Autorité, Prénoms, délivrance, and spéciales), and above the third line (which should read "Given Names / Prénoms / Appelidos"), the word Prénom is not capitalized. Finally, at the bottom, where the full name is repeated for computer scan, it shows P<USAGABRIELLE<<ASHDOWN<<GABRIELLE<<<<<<<<<<<. However, real American passports have surname listed first, followed by given names (separted by only one < and not two), so it should display as P<USAASHDOWN<<GABRIELLE<<<<<<<<<<<. This passport would read as if her name was "Ashdown Gabrielle Gabrielle" and would not match the other information on the passport. No skilled forger would make so many mistakes, and if she had acquired an authentic passport using a stolen identity, it would be correct.
It is highly unlikely that one policeman would barge into a room where two shooters are and without any warning, shoot only one of them. Then, like nothing happened, the story continues.
It's nearly impossible that the dead body of the young man in the car could go unnoticed for a week. Even in a closed-up vehicle (which would not be airtight), the horrible and unmistakable stench of decomposition would be obvious throughout a large radius within a few days, and would be unbearable even inside the house (and possible neighboring homes) well before a week had passed.
(50:48) When Sherlock is talking to Mycroft the boom mic is visible bobbing above Sherlock's head.
As a doctor John Watson would have immediately placed Mary flat on the ground and placed pressure on the wound. He would not have just left her propped at an awkward angle and done nothing as she bled out.
Through out this episode Mary and John are both scene holding handguns in defensive positions. John being a former military man and Mary being a former highly trained WET agent display poor handling of the firearms.
I.e Both index fingers on the trigger or overlapping each other. Poor trigger control, as in having the finger on the trigger when there is no imminent threat.