Nice thread.
Back in the 60's the Morman Tabernacle Choir had a Top 10 hit with the "Battle Hymn of the Republic" (If I'm lyin', I'm dyin').
The Benny Goodman Band, featuring Ziggy Elman, covered a classic Klezmer tune as "And The Angels Sing". Also, from the big band era, Ella Fitzgerald, with the Chic Webb Band, doing "A Tisket, A Tasket".
In the 50's, several popular Italian-American artists (including Dean Martin and Perry Como) covered O Sole Mio as There;s No Tomorrow.
Traditional songs have been a source for pop writers for ever.