Show of Hands play a cover of "Time after time."
They also cover "First we take Manhattan", as do many other people. That song itself is in a somewhat dodgy category, because Leonard Cohen is fairly well recognised in the folk world, but that album was more of a rocked-up chanson kind of thing, and he's since become rather too keen on Casio keyboards and cheap drum machines.
And conversely, whilst Paul Weller is best known for his electric stuff, "Wild Wood" is a very folkie song. Rather like Dire Straits' "Why worry" and "Romeo and Juliet" which crop up periodically at folk clubs.