Several years earlier than the Bridge & the Topic, tho, there were the 'skiffle' clubs ~~ a sort of jazzy/folky genre which involved impro'd household implements like washboard played with thimbles for the percussive element & bass made from a teachest & string ~~ see the very informative Wikipdedia article on "Skiffle". It is hard always to follow directly as to 'what grew out of which?'; but there is surely no doubt that the origins of the folk club as we know it were intimately bound up with the brief but influential 'skiffle' craze of the early-mid 50s ~~ many who liked folk were interested by the skiffle phenomenon, & folk songs might well be heard during the 'open' part of a skiffle evening.
Note particularly the following from wiki --
A large number of British musicians began their careers playing skiffle in this period, and some became leading figures in their respective fields. These included leading Northern Irish musician Van Morrison and British blues pioneer Alexis Korner, as well as Ronnie Wood, Alex Harvey and Mick Jagger; folk musicians Martin Carthy, John Renbourn and Ashley Hutchings; rock musicians Roger Daltrey, Jimmy Page, Ritchie Blackmore, Robin Trower and David Gilmour; and popular beat-music successes Graham Nash and Allan Clarke of The Hollies. Most notably, The Beatles evolved from John Lennon's skiffle group The Quarrymen.

~M~