Being a British-based series of stories, the Sherlock Holmes Canon doesn't mention presidents much. In fact, there's only one use of the word "president" in that whole mass of verbiage:
"... this period includes the case of the papers of Ex-President Murillo ..."
Unlike many another such mention, that line in "Norwood Builder" actually refers to another case that we've actually read. But like a few other mentions, this reference doesn't exactly line up with the facts were were given in that story.
The papers of Ex-President Murillo? What papers? There aren't any papers of note in "The Adventure of the Wisteria Lodge."
Don Juan Murillo, the dictatorial president of the Central American country of San Murillo, was known for his cruelty and his ongoing flight from punishment resulting from that cruelty. He fled to Paris, Rome, Madrid, Barcelona, England, and when his past started to catch up to him, he fled to Spain, and was finally murdered by his enemies.
Don Juan Murillo is one of those folks whose path never actually crossed that of Sherlock Holmes, one of those villains whose legacy actually just spawned other crime and mystery in its aftershocks for Holmes to investigate. We never knew the full extent of his crimes, but they were bad enough that his name alone immediately causes Watson to recognize him as "the most lewd and bloodthirsty tyrant that had ever governed any country with a pretense to civilization."
Don't mistake any of this as a definite parallel to a certain ex-president we wave good-bye to today in America. Murillo was really good a being a villain:
"Strong, fearless, and energetic, he had sufficient virtue to enable him to impose his odious vices upon a cowering people for ten or twelve years," Watson wrote. "As cunning as he was cruel, and at the first whisper of coming trouble, he secretly conveyed his treasures aboard a ship with was manned by devoted adherents."
Murillo milked his country dry for about a dozen years and then ran off with the treasures he piled up after he had gotten so bad that "there was a universal uprising against him." Murillo plainly didn't have an effective bunch of disinformation networks and social media working for him in San Pedro, but still made it over a decade in his dictatorship presidency.
We never hear what happened in San Pedro when they were finally free of the bastard Murillo. Sherlock Holmes refers to it with "the backwoods of San Pedro," but did he mean the whole country or just its actual backwoods regions?
Hard to say. But they were certainly rid of Don Murillo. And I'm sure they were very happy about that.