This is what Pete himself writes in 'Where Have All the Flowers Gone':

[1993:] In the late '50's I got a letter: "Dear Pete Seeger: I've made what I think is a singable translation of a poem by the Turkish poet, Nazim Hikmet. Do you think you could make a tune for it? (Signed), Jeanette Turner." I tried for a week. Failed. Meanwhile I couldn't get out of my head an extraordinary melody put together by an Massachusetts Institute of Technology student who had put a new tune to a mystical ballad The Great Silkie from the Shetland Islands north of Scotland. Without his permission I used his melody for Hikmet's words. It was wrong of me. I should have gotten his permission. But it worked. The Byrds made a good recording of it, electric guitars and all. [...] I never met Jeanette Turner, who was a volunteer with a New York peace organization. She died soon after she wrote me. Bless your memory, Jeanette. And Hikmet, the Turkish Communist poet imprisoned for so many years. And thank you, James Waters, now a professor in Vermont. Your melody is one of the world's greatest. (Seeger, Flowers 195f)

So the lyrics should be credited to Hikmet, transl. by Turner / Seeger, the tune to Waters. I think (can't check just now) the credits in the book leave out Turner.

It's one of my all-time favourite songs.