This particular series takes the characters and puts them in present day London, what was normal in the 19th century doesn't apply for the character in the example

The majority of the UK had guns readily available up until the great war.

[sic] there were no firearm restrictions in effect, but less than 1% of the population had firearms. The Firearms act of 1920 disarmed this percentage over fears of surges in crime, as well as worries that the working class would get too unruly

Sherlock is also armed in several of the original sir. Arthur Conan Doyle stories. Iirc most the time he borrows a revolver from Scotland Yard.

I have no knowledge about the UK, but I can say that was the case for Germany. Gun control in the modern sense only began after WW1 when large numbers of weapons fell into civilian hands.

In the 19th century, shooting practice was mandatory as part of the military system; reserve and conscription. Kinda like how it still survives in Switzerland. But as the UK rarely had a need for large land armies, the population was not militarized to such a degree.

I think that everyone is over complicating this.

When you see he has a gun in the mini-series, you're just reminded of how the he had a revolver in the books. You aren't going to think about it because he's supposed to have a gun. You just say, "Oh, he still has a gun."

I think the disconnect is probably between people who know Dr. Watson from the books, and people who don't know the books. The Sherlock mini-series is obviously targeted at people who never read the books. So it's entirely possible that they wanted the viewer to question its existence.