Love him or hate him, there's one thing you can't deny about Steven Moffat - the co-creator of the BBC's smash hit Sherlock and long running showrunner (but soon to be usurped) of the BBC's other smash hit show Doctor Who - and that's the fact that the man really knows how to work his fans.
As two of the most popular shows to ever come out of the modern BBC Sherlock and Doctor Who have achieved international reach, rooting themselves in American consciousness as good examples of fine British broadcasting. The two also boast highly active and endlessly creative fanbases, which is great (most of the time anyway).
In this way they've become almost representative of the UK television universe within the US, and I completely understand where people are coming from when they envision the scope for a crossover between the two.
There's a metric shit-ton of fan-made crossover videos, crossover art, crossover literature and other media on that theme floating around the internet (though be careful not to dive too deep incase you see something you really didn't want to - so much sexual tension).
Never one to shy away from whipping up the fans, at a panel back in July 2015 Moffat was asked the question as to whether or not we'd ever see an official crossover between the two shows. And he gave a fairly positive response towards the idea:
"There must be some sodding panel I can go to... [laughs]. Hey listen, talk to all the rest. My wife [Sue Vertue], [Mark] Gatiss, [Benedict] Cumberbatch, [Martin] Freeman - I'm fine with it. I'm the nice one! It's the rest of them... You're assuming I have some power over it!"
Understandably this excited a few fans, proponents pointing to this exchange as proof of the fact that Moffat himself would be open to the idea of doing a Sherlock / Doctor Who crossover; some even saying this was a sign that Moffat "totally wants to" make this happen. But this prospect also worried some fans, even a non canonised version of the shows would still be... a little odd, shall we say.
But Moffat backtracked on his comments in an interview with Collider a few months later when he was again asked the million dollar question lauded by fans of the two shows. The results are in, brace yourself:
"That’s a question that I get asked so often, and I can’t keep answering it. It’s all right for Doctor Who. That’s fine. But it would change Sherlock’s life, if he met the Doctor and knew that time travel was possible. He’d have to factor that into every crime he solved. And do we really think that Sherlock Holmes lived through a Dalek invasion? I don’t think he did. I think he’d have mentioned it by now. It’s not going to happen. That’s just the truth of it."
There you have it.
To be honest we doubt anyone ever really expected that this in-joke was going to come to fruition (especially given how difficult it is just to get one episode of Sherlock made nowadays) but the point Moffat makes is that the concept doesn't translate well back to the source material, Sherlock in particular. It's a good point to make, and one which we fully agree with him upon. Even looking past the fact that the two narrative universes wouldn't fit neatly into each other the main characters themselves are also pretty incompatible.
Both the Doctor and Sherlock are the centrepieces of their respective shows, which you can tell even just by looking at the series' names. Their characters are the central factor around which each narrative rotates; putting the two of them together would be too much, as Moffat himself put it in the Collider interview:
"It’s a bit like when the Doctors meet [in The Day of the Doctor]. It’s fantastic, at the beginning, and then you think, 'What the hell am I going to do with three of the bastards?' It’s just weird. You don’t really need more than one person like that in a show."
I think people just really want to see Sherlock fight a Dalek, and we're actually absolutely on board with that.
But even though it's never going to happen there's still nothing wrong with imagining this crossover through the wonderful medium of internet mash-up culture. Most of the time it's a lot of harmless fun for fans to enjoy and explore alternate readings of their favourite shows, and there's nothing wrong with that (but then sometimes we get things like Twilight fan-fiction becoming 50 Shades of Grey and we start rethinking our stance on that).
Some of the crossover fan-art which has been created by fans thus far is really great, largely because they're little snatches into this amusing, subjective universe. But at the end of the day that's all they are, funny little fictional anecdotes.
The closest we're probably ever going to get to an official Who / Sherlock crossover is the Seventh Doctor (Sylvester McCoy) novel All-Consuming Fire, published back in 1994, in which the Doctor teams up with a classical Conan Doyle-version of Sherlock Holmes to defeat the god Azathoth.
The Doctor and Sherlock battling ancient gods, what more can you want? And if that's not enough, there's always the fan versions, which Moffat is in full support of.