WNU-TOOB: THE DIOGENES CLUB

Although the televersion of the Sherlock Holmes story "The Adventure Of The Greek Interpreter" veered wildly from the original (thus proving that Toobworld's Holmes is not the same as the literary version), one thing remained the same - the Diogenes Club, that sanctuary for the odd gentlemen of London where they might seal themselves away from the distractions of the world.
Here's how it's described in Wikipedia, with the appropriate quote from the Dr. Watson manuscript (edited by Conan Doyle):

The Diogenes Club is a fictional gentleman's club created by Sir Arthur Conan Doyle and featured in several Sherlock Holmes stories, most notably "The Greek Interpreter". It seems to have been named after Diogenes the Cynic (although this is never explained in the original stories) and was co-founded by Sherlock's indolent older brother, Mycroft Holmes.

The club is described by Sherlock Holmes in the stories thus:

"There are many men in London, you know, who, some from shyness, some from misanthropy, have no wish for the company of their fellows. Yet they are not averse to comfortable chairs and the latest periodicals. It is for the convenience of these that the Diogenes Club was started, and it now contains the most unsociable and unclubable men in town. No member is permitted to take the least notice of any other one. Save in the Stranger's Room, no talking is, under any circumstances, allowed, and three offences, if brought to the notice of the committee, render the talker liable to expulsion. My brother was one of the founders, and I have myself found it a very soothing atmosphere."

– The Greek Interpreter

It is described as a place where men can go to read without any distractions, and as such the number one rule is that there is no talking, to the point where club members can be excluded for coughing.

In the Wold Newton Universe, triggered by its appearance in the movie "The Private Life Of Sherlock Holmes", the Diogenes Club also serves as a front for the British Secret Service and apparently is the center of attention for the investigation into the occult. (I imagine the Torchwood Institute remained the hub for the containment of alien activity in the British Empire.)

Toobworld Central accepts this premise, as there is nothing to invalidate it (so far as I know) in other TV presentations. In fact, the suggestion could create links to other TV shows and TV movies that might otherwise still be drifting free in the Great Link.

For example:'Colonel March of Scotland Yard' was likely a member of the Diogenes Club, as a cover for his Department of Queer Complaints. William Sebastian, who faced the "Spectre" of his own mortality because of his opposition to the Dark Powers, may have come to the club for advice in his investigations. And almost certainly Sebastian was probably ordered there to be debriefed at the club by British Intelligence after the Cyon case in London back in 1977.
But in order to maintain that cover, the Diogenes Club still had to operate as a gentlemen's club. And as such, I think we have found another TV show - another in the classic detective genre - where the Diogenes Club made an appearance.
After a day of sight-seeing throughout London, Lieutenant Columbo was brought to the club where his host, Detective Chief Superintendent William Durk, was a member. There they were to enjoy "tea", which 'Columbo' thought would consist of those dainty sandwiches. Instead he was treated a traveling sideboard of hearty fare - fruits, roasted beef, cheeses of all types - to this day I wish I could take a crack at sampling the options when I watch that scene.
Unfortunately for Columbo, the pathologist Divers showed up with unsettling autopsy photos that put the Lieutenant off his feed.

Apparently this scene in the episode "Dagger Of The Mind" takes place in the Strangers' Room of the club, which has seen much embellishment since its days when Mycroft Holmes co-founded the place. Since Durk is a member, it's quite possible that he is somehow connected to the Secret Service. He may even have been mentored by Colonel Percival March.

[This is NOT a picture of them working undercover!]

As to the differences in the club entrances seen in "Dagger Of The Mind" and "The Adventure Of The Greek Interpreter", it's simply a matter of them being on different streets as the club has expanded to take up the entire block (although it's not visibly apparent from the outside that the various buildings all inter-connected, even though they are of different architectural styles.
The entrance seen in 'Columbo' would be the unassuming public entrance for the regular members who have no clue of what else goes on there in the club. And the more imposing doorway down a side street which was used by 'Sherlock Holmes' and the first Dr. Watson* has now become the highly guarded entry for those in the Secret Service.
O'Bviously this is a "missing link", but one that can provide common ground with the alternate world of the Wold Newton Universe, even if the characters from the Sherlock Holmes stories differ.

TV SHOWS & TV MOVIES CITED:
'The Adventures Of Sherlock Holmes'
'Columbo'
'Colonel March Of Scotland Yard'
"Spectre"
'The Girl From U.N.C.L.E.'

BCnU!