These existed in public houses from Victorian(?) times and have been described as a poor man's freemasonary. The Cider Bar in Newton Abbot has a club still in existence and I have references to clubs in Greenwich and Rushden.
A quick Google confirms: the Scrumpy User Guide entry for Ye Olde Cider Bar has brief mention and a picture of the Long Bar Cork Club that last met in 1911; the Rushden Research Group has an account of how Rushden Cork Clubs worked; and there's similar at the Greenwich Phantom (May I See Your Cork, Brother?). The "cork" referred to the insignia of membership - a brass-bound cork - and the charitable aspects - usually ploughed back into the club to fund an outing - were funded by levying fines for swearing (helped along by the general disinhibition of the clubs' drinking sessions). Perhaps this disreputable angle is what makes information on these clubs hard to find; perhaps the general invisibility of working-class leisure and social life in the mainstream record. Anyhow, I was able to track down just two references in The Times:
"THE MEANING OF A CORK CLUB"
The mention that a cork club was held at a public house led Mr Wallace to ask, "Has it anything to do with cork?"
Mr George Elliott, K.C. (for the licensee)— I believe not. I understand that it is a harmless holiday club, but it is just possible that corks may be remotely connected with the holiday.
Mr Bodkin, who opposed, said that the members of such a club competed in collecting corks, the rule being that the corks must have been drawn from bottles, the contents of which had been consumed by the competitors. (Laughter.)
Mr. Elliott.— I have never envied my friend's brilliant imagination more than now. (Laughter.)
- page 3, The Times, Jun 20, 1912
There was further discussion at the licensing meeting of the London justices at the sessions house, Newington, yesterday, as to what constitutes a "Cork Club."
Mr Wootton, representing a licensee, said that the clubs had strict rules. At their meetings a cork was placed at the centre of the table, and if there was any breach of the rules the offender was fined a penny or any sum agreed upon, the fines being afterward spent on a holiday.
Sir Douglas Straight, one of the justices, said that he understood that a cork club meant that members were compelled to carry a cork wherever they went, and were called upon to "stand" drinks all round if challenged when they had no cork in their possession.
- page 3, The Times, Jun 21, 1912.
One of the online examples, the Cork Club of the Bull's Head, 39 Market St. Oakengates (see here) has a slightly different name, "Merrington's Jolly Corks".
I don't know if this implies any connection with a similar organisation in the USA around half a century before, the Jolly Corks. As described at Charles Vivian & The Jolly Corks, this was originally founded in New York City in 1867 by a group of musicians and entertainers, who organised private variety shows to get around the "blue laws" forbidding Sunday drinking. They took their name from a "cork trick", a semantic dodge to con someone into buying a round, and, like the later English cork clubs, required members to carry a cork. They rapidly affiliated on a more formal basis as a benevolent organisation, rebadged in 1868 as the Benevolent and Protective Order of Elks, which still exists as one of the the leading US fraternal organisations.
There's a minor local connection. As described in A biographical sketch of the life of Charles Algernon Sidney Vivian, founder of the order of Elks : together with anecdotes and reminiscences of his work and travels (Imogen Holbrook Vivian, 1903, Internet Archive ID biographicalsket00vivirich), the main founder was an English émigré born in Exeter in 1846, and the son of "a clergyman of the Established Church". The biography says:
He never tired in describing to me the long walks by the sea he used to take in boyhood days by his father's side, near Exeter, in fair Devonshire, always alluding to him in the most affectionate terms, with fond remembrance of those delightful hours spent in pleasant and instructive conversation as they walked the sea-girt shores of old England.
I haven't yet tracked who his father might be; some accounts say "Vivian" was a stage name, and that Vivian's real name was Charlie Robertson. Accounts also differ as to his birth date.
Addendum: I just found another reference, showing that the English cork clubs weren't necessarily all harmless bonhomie.
Among the many new clubs that have been of late established, few possess the advantages to members offered by a club at Runcorn in Cheshire, called the "Cork Club." This club, it is stated, is at present attracting much attention in the town, owing to the exuberant spirits of its members, who are bonded together by fraternal ties, and by the rules of the club are bound to assist each other in time of trouble. Their fellowship, however, is carried almost too far for the comfort of such of the inhabitants of Runcorn as have not the honour of belonging to the club, and recent cases before the magistrates have elicited the information that the members of the club are in the habit of surrounding unsuspecting persons and chastising them severely, doubtless not without reason, but still more severely than is technically justified by law. Several of the members enter a public-house, one of them picks a quarrel with a stranger, and a fight is arranged. If the member of the Cork Club is getting worsted, his fellow members join in the fray. The police are therefore making a raid at present on the club, owing to the number of serious assaults reported as having been committed by its members, and on Tuesday the chairman of the club was committed to prison for a month, having assaulted a volunteer in the drill-hall. This, of course, casts a gloom for the moment over the club; but the month will soon pass away, and the chairman be at liberty again. The token of membership of this interesting fraternity is the possession of a cork, and any member found to be minus the cork is compelled to "stand drinks" for the person who makes the discovery.
- The Pall Mall Gazette, December 17, 1874
Ah, the good old days ...