Martha Hudson does not exist in the Canon. Because of Vincent Starrett's essay "The Singular Adventure of Martha Hudson," Sherlockians have taken to heart that an imaginary character can be created from three very different ones: a reliable landlady, a gossipy housekeeper, and a quiet spy.
The name 'Martha' is only used in "His Last Bow." She is the only servant left at Von Bork's mansion on the night that the story takes place. While the German agent is showing off to his comrade, Baron Von Herling, the dear old ruddy-faced woman, accompanied by her cat, was bent over her knitting in a different room. As far as Von Bork was concerned, the old lady's self absorption and sleepiness personified Britannia.
We later learn that Martha has been placed in Von Bork's mansion by Holmes himself and that night signaled Holmes and Watson to arrive at the mansion only after the Baron has left when Holmes could capture the spy without incident. This spy cannot be the Baker Street landlady.
Sure, Mrs. Hudson was helpful by turning the wax dummy of Holmes in "The Empty House," but the woman who had to be instructed to crawl so she wouldn't be seen is now the one privy to Holmes's plans and is making judgments on whether or not he should arrive with the Baron in the house? Mrs. Hudson, a woman who was excited to find a revolver bullet in the carpet of Baker Street has grown into someone who monitors letters written by spies? No, it will hardly do.
"But," some would say, "maybe the landlady grew tired of her digs in London and wanted a life of adventure?"
Sure, people long for adventure, but as D. Martin Dakin points out in A Sherlock Holmes Commentary, this theory will not hold water merely for the fact that Holmes addresses the woman in LAST by her first name. "Victorian formality of speech ruled out the use of Christian names for anyone but members of the family (like Mycroft), children (like Jack Ferguson) or servants (like Billy the page or John in 'The Man With the Twisted Lip'). Watson was Holmes's most intimate friend, to whom he was deeply devoted, yet never in all their years together does he call him anything but 'Watson', except when he is 'Doctor' or some similar phrase. It is incredible that he would have been so guilty of such familiarity with the landlady whom, we are told, he had always treated with such gentleness and courtesy."
But the Mandela effect can be strong, so let me offer one more piece of evidence to prove that 'Martha' is not 'Mrs. Hudson.' At the end of EMPT, Holmes is checking in with Mrs. Hudson and shows concern for her well being (“I hope you preserved all precautions, Mrs. Hudson?”), praise for a small task (“Excellent. You carried the thing out very well."), and then a curt dismissal as she is no longer needed for the case at hand ("All right, Mrs. Hudson, I am much obliged for your assistance.").
Holmes's interactions with Martha are quite different. He describes her to Watson as someone "who has played her part to admiration." When she arrives, Holmes says, "Ah, Martha, you will be glad to hear that all is well.” She shows knowledge of the mission as she tells Holmes that she could not have left the previous night when Von Bork wanted her to accompany his wife, because "that would hardly have suited your plans." And, although Holmes dismisses her in his typically curt manner, he makes arrangements to meet with her the following day at Claridge's Hotel back in London. No, Martha was not an old friend whom Holmes asked to help him out. This was a trained agent that Sherlock Holmes trusted to carry out a role and to make decisions that would allow him to achieve his goal of capturing a German spy.
So let go of those fantasies of the nice landlady that doubles as an international spy. Martha and Mrs. Hudson are two different people. Plus, Sherlock Holmes's landlady never owned a cat.