I think the key word is Club as in a Membership, a Committee, Rules, a Mission statement and an annual General Meeting and Elections. My understanding is that in 1956 when the entire Bradford Communist Party resigned in protest, when the Russians sent in their tanks to Hungary, they then founded the Topic Folk Club. To enable them to still meet on a Friday Night and sing the same songs. I would think that The Topic Folk Club was in all probability meeting the criteria of being a club from the start. The night that it meets and its mission statement may have changed over the years but a Club it is and a Club may it long remain.

Some Folk artists were allegedly sponsored by the Communist Party to promote communist ideals. This you must remember was a very different world and not always for the better. One American Folk Singer had visited Communist China and was not allowed back into the United States of America known then and now as the land of the free.

It matters not whether you agree or disagree with politics of a song what is important is that the other view's should be heard. To that end in theory Folk Clubs have not done a bad job enabling ordinary people to air their views.

It's my understanding that some of the Folk Venues in London have to have a membership to enable a Folk Night to take place in a particular venue. The change of venue for any number of reasons that may or not have meant a new membership and change of name even if the core of the club had not changed it would be a New Club.
The Grove Folk Club in Leeds, Aren't we lucky having two historic Folk Clubs within a few miles of each other, was established 1963 as the venue and the Club have been so much a part of the lives of it's regulars. That it should claim to be the oldest established Folk Club in the Same Venue The Grove Inn, Back Row, Holbeck, Leeds just stones throw from Leeds City Centre.