I think it depends a bit what you call "country" - country/roots/Americana (think Doc Watson, Peggy Seeger, Carter Family, Bluegrass at a stretch) did crop up in folk clubs / folk singer repertoires to a degree in those times; more "commercial country" (think 40s/50s/60s tastes) generally not, in my experience at least, except perhaps that U.S. singer songwriter material (Tom Paxton as mentioned earlier, others later such as Steve Goodman and John Prine) certainly did, although I would really consider these more at the "folk" end of the country spectrum, if they belong there at all.

Some of the "new country" 70s/80s acts and singer songwriters (Emmylou, Peter Rowan, Tony Rice etc.) also appealed to my more "folky" (but fairly catholic) tastes of the day but in general would have been too "big" by then to have played the folk club circuit, although the odd one might be included in a "folk" festival lineup.

Speaking of the latter, the Cambridge Folk Festival was the main one of those I attended to have a bit of a home for "country" (and country-ish) local and U.S. acts - possibly Nick Barraclough's influence?? - as Festival programmes of the day will attest, so maybe there was a bit more cross fertilisation in the clubs in that area - others will know better than me.

Check out Cambridge Folk Festival: Past Artists for a full listing of these.