Just so I’m clear, is it that actors change characters, but within the same universe, and with a more prominent role? I’m a little confused by the replies.
The “pure” examples would be shows/films with a specific main character played by X, and with either a “sidekick” or “other minor character” played by Y, and then in a later show/film involving the same character, Y plays the main character instead. In OP’s example you had someone literally who played Watson (Holmes’ famous sidekick) in a later version play Holmes himself.
It would seem there’s variations on this theme though, depending on how much you care to explore those.
For example Martin Sheen played the Chief of Staff to Michael Douglas’ POTUS in The American President, a few years later Martin was playing the President in The West Wing, that wouldn’t be a perfect example because while Chief of Staff is arguably Number Two to POTUS, the actual characters in The American President didn’t carryover to The West Wing–even though I do think the West Wing has been said to have been partially inspired by The American President–and Martin’s casting as President Bartlett specifically desired because of his performance in the previous film–you’ll note Aaron Sorkin wrote both works.
Maybe a convoluted example–Larry David, who had a moderate career as a stand-up comedian, sketch show performer and writer prior to Seinfeld–but no real experience as an actor, was the executive producer, writer of that show. He appeared in that show numerous times, sometimes with voice-only characters, sometimes on screen but only briefly (as the enigmatic “lawyer” in one episode, as a convenience store clerk in another etc.) However, the character of George Costanza (portrayed by Jason Alexander) was essentially based on Larry David. Years after Seinfeld went off the air Larry starred in Curb Your Enthusiasm…as a fictionalized version of himself–this time played by himself and not Jason Alexander.
Doctor Who did this quite a bit.
Colin Baker played a small role as Commander Maxil in Arc of Infinity before being cast at the Doctor.
Ian Marter was a ship’s officer in “Carnival of Monsters” and later played the companion Harry Sullivan.
Karen Gillan played a soothsayer in “The Fires of Pompeii” and became Amy Pond later.
As always happens, people either don’t understand the OP concept or try to broaden things. Several answers here have veered considerably from what I’m looking for, which is cases within a film series or TV series (or at least cases based on the same book or whatever) where someone who originally played one character later got elevated into the lead character. It might be a different series (as with David Suchet playing Poirot in a series unconnected with the one in which Peter Ustinov played Poirot, but Suchet had appeared in as Japp.
That didn’t happen with Ma and Pa Kettle, for instance – that’s a different type of thing, where a side character ends up with their own spinoff or their own series. Ditto The Fonz. Henry Winkler didn’t play any other character besides Arthur Fonzarelli – his character simply gained in importance. Not the same thing.
Another example from James Bond movies, Robert Brown first appeared in TSWLM as Admiral Hargreaves, a minor character. He later played the much more significant role of the second M.
There has been lots of debate among Bond fans if Hargreaves took over the job, but they are probably intended to be separate characters.
Tim Russ played minor characters in multiple Star Trek series – “Devor” in TNG and “T’Kar” in DS9, before being cast as Tuvok in Voyager.
IIRC, Billy de Wolfe was the station manager. I can’t remember who the other DJ was without looking it up.
Turns out it was Joby Baker. Who apparently found steady work as an actor, though I don’t remember him from anything else.
Several better known people were in the cast. I laughed when I saw Goldie’s character was named “Kramer.”
Despite having the same creator, Petticoat Junction was not, strictly speaking, a spin-off of The Beverly Hillbillies. However during season 6 of Junction, it was established that Cousin Pearl Beaudine of Hillbillies (played by Bea Benaderet) was in some way related to Kate Bradley of Junction (also Benaderet.) In several crossover episodes, the cast of Hillbillies interacted with Benaderet as Kate Bradley, and not as Cousin Pearl.
I agree. It was better without his self-conscious actorliness and her oh-so-smouldering looks sans any acting ability whatever.
You must have read the satire of it in MAD Magazine. Once I had, I couldn’t watch the show without laughing each time they showed close-ups of those two. :My favorite episode is the one where they do the cabaret act in “Ramsdorf, East Europe.”
If you drink … my glass … of wine … 
Mort Drucker’s caricatures of Greg Morris were especially hilarious too!
Bain, however, was really hot as Rob Petrie’s old girlfriend on The Dick van Dyke Show. Hubba-hubba! 
Patrick McNee not only played Dr Watson [in Sherlock Holmes in New York (TVM 1976), Sherlock Holmes and the Leading Lady (TVM 1991), Incident at Victoria Falls (TVM 1992)] and Sherlock Holmes himself [in The Hound of London (TVM 1993)] but also someone who thought he was Sherlock Holmes [in Magnum PI: Holmes Is Where the Heart Is (1984)].
Here’s one that goes in both directions:
Jeff Conaway played the lead Danny Zuko for over 2 years in the Broadway production of Grease. For the film version, he played Zuko’s sidekick Kenickie.
Meanwhile, John Travolta played the even more minor sidekick Doody on Broadway, but ended up in the lead for the film version.
Here’s one that should fit the OP’s criteria exactly.
Martin Sheen played Robert. F Kennedy in the 1974 TV movie The Missiles of October In 1983 he played John F. Kennedy in the miniseries Kennedy.
Sara Gilbert got quite a promotion from Roseanne to The Conners. Sean Murray and Brian Dietzen have, on NCIS, risen from comic relief butt-monkeys to, in the current season, arguably the two most important characters on the show. Robert Duncan MacNeil went from a minor character in ST:TNG to a very important character (with an almost identical backstory) on ST:Voyager. On Mad Men, Kiernan Shipka’s character, Sally Draper, started off being no more important than brother Bobby. Bobby became increasingly unimportant (No one noticed that a string of different actors played him until he made a joke about it) and Sally had major episodes built around her by the final season.
Rob Lowe, Allyson Janney and Richard Schiff’s characters were all on about the same footing in season one of The West Wing, but were on wildly different trajectories by the final episode.
I think this counts. Lucy Lawless had a few minor roles as different characters in the TV show Hercules: The Legendary Journeys before getting the role of Xena, first on Hercules and then on the spin off show Xena: Warrior Princess.
Adding to the trivia. Her biggest non-Xena role on Hercules was as Lyla, a woman who married a centaur. She portrayed Lyla both before and then after she had become established as Xena leading to some fourth wall breaking, eye winking as other characters discuss Lyla saying ‘Does she remind you of someone…?’
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Michael Caine got an Oscar nomination for expertly second-banana-ing Laurence Olivier in SLEUTH, playing Milo Tindle to the other guy’s Andrew Wyke; and, after Caine won some Oscars of his own, he first-banana-ed Jude Law, as the Wyke to his Tindle.
I considered this, but I’m not sure there really is a “second banana” in what is essentially a two-character film.
To be honest both roles are pretty much equal status but Johnny Lee Miller and Benedict Cumberbatch (both did notable modern versions of Sherlock Holmes) did a stage production of Frankenstein where they alternated between portraying Dr Frankenstein and his creation. The plays were filmed and both versions were released.
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