There is no guarantee that the TV characters of Earth Prime-Time will have doppelgangers in alternate dimensions. A case in point, and keeping it within the 'Doctor Who' framework, is that of the Rose Tyler from the Earth where John Lumic built his own version of Cybermen. On that world, Rose was never even born. For Pete and Jackie Tyler, the only Rose in their life was not a daughter but a dog.
In some TV dimensions, there would be characters who may have been born into the alternate worlds, but who died before they had the chance to live out their destinies as counterparts or variants to the originals. And the same would be true of those literary characters who were adapted for television, including Sherlock Holmes.
There have been many adaptations of the Great Detective on TV - several in full series, but most in one-off TV movies, and not all of them adapted from the official canon. There are plenty of Sherlocks to go around - from Jeremy Brett (the Holmes of Earth Prime-Time) to John Cleese and Peter Cushing to Roger Moore and Douglas Wilmer, to Leslie Howard's son and a 22nd century cyborg to be found in the Tooniverse. And now we're about to be graced with the second season of 'Sherlock' with Benedict Cumberbatch as the spirit of the original Holmes now existing in that 'Life On Mars' Limbo. (That's the Toobworld Central splainin, folks.)
But what if Holmes died off in one of those alternate TV dimensions before he ever teamed up with Dr. Watson? Not much is known about the early life of Sherlock Holmes. There are theories that his parents died tragically in a fire or that they abandoned young Sherlock and his brother Mycroft. And "The Seven Percent Solution" put forth the claim that Professor Moriarty was involved in the fates of his parents.
So what if tragedy befell young Sherlock and he died as well?
Any world without Sherlock Holmes would be a poorer place, but it would also be a dangerous one if Professor James Moriarty was allowed to run unchecked. As the author of "The Dynamics Of An Asteroid", could the Napoleon of Crime eventually have brought an aline invasion to Earth sooner than it could be prepared for that eventuality?
This being a 'Doctor Who' marathon, it's about time we brought the Time Lord into this discussion.
In an alternate TV dimension in which there was no Sherlock Holmes, that version of the Doctor would have no clue that something was amiss, that the Game could never be afoot. It would take the Doctor visiting from another dimension to notice Holmes' absence, having read Dr. Watson's transcripts of their adventures, perhaps even having met him while in the Victorian past.
(This is one of those times when I wish the Toobworld Dynamic could be more like shared-universe associates the WNU and the TVCNU which also incorporate stories and characters from printed source materials. There's a fine novel entitled "The All-Consuming Fire" which teamed Holmes and Watson with the Seventh Incarnation of the Doctor. It would have made for an interesting two-parter episode for the TV series.)
So let's say the Doctor of Earth Prime-Time visited an alternate TV dimension in which Sherlock Holmes didn't exist. This would be during that time when the other Time Lords were still around, when travel between dimensions was still easily achieved.
Realizing how much danger the Earth of that dimension would be in without a Sherlock Holmes, the Doctor would probably undertake the only logical option - he would stay in that dimension and become its Sherlock Holmes.
(To my way of thinking, this would have to occur after the Fourth Doctor's return from Gallifrey but before he picked up Leela as his companion.)
For whatever reason, the Doctor would have arrived on that Holmes-less Earth around 1881, just in time to take part in that first contact with Dr. Watson. Being a Time Lord, there would have been no problem if he stayed several years in that other dimension. With the TARDIS he could return to his home dimension within moments after his departure.
I think the Doctor would have happily settled into the life that Sherlock Holmes never got the chance to live, with all the trappings - the Persian slipper full of tobacco, the deerstalker cap and the Inverness cape, the initials "V.R." made in the wall with gun shots. (For the Doctor, "V.R." might have meant something other than "Victoria Regina".)
And with his foreknowledge of the cases recorded by his "Boswell", the Doctor would have been able to solve every case in record time, perhaps sometimes before the client ever came to call on him at 221 B Baker Street.
I could see the Doctor staying long enough to play out the rivalry with Professor Moriarty to its pre-ordained conclusion at the Reichenbach Falls. (And maybe with an alien sub-plot to sweeten the mix?) Afterwords, he would have departed back to Earth Prime-Time, leaving those other cases - from "The Empty House" to "His Last Bow" - to be solved by those detectives and other investigative agents who were considered 'The Rivals Of Sherlock Holmes'.
When the Doctor returned to Earth Prime-Time, he took the whole wardrobe he had assembled for his Holmes persona. It would later come in handy when he and Leela investigated the case which would be known as "The Talons Of Weng Chiang".