When Britten rewrote The Lyke Wake Dirge for Peter Pears, it lost its 'folk' form and became something else

He didn't rewrite it. His setting is from 1943. This documents the Young Tradition version, which is probably the one you're thinking of: seems to have been created by Hans Fried in the early 1960s.

https://mainlynorfolk.info/peter.bellamy/songs/lykewakedirge.html

More here on Ian Pittaway's site:

https://earlymusicmuse.com/lyke-wake-dirge/

which also includes an unrelated tune written down in 1929. That makes three other tunes that don't sound anything like Britten's. I'd say Britten's sounds as folky as any of them; I could easily imagine it from a group of pub singers with the refrain lines done in chorus.