The DC-area group Clam Chowder does a wonderful version of the medieval Deo Gratias Anglia to the tune of the Banana Boat Song. They sing one verse straight first --
Our King went forth to Normandy
With faith and might of chivalrie
Therefore England doth call and crie
Deo Gratias Anglia
before the tenor breaks out into Belafonte's
Deo gratias Anglia
The calypso rhythms start, and it all goes downhill from there.
In Sacred Harp music, there's the concept of Common Meter -- 4/4 time with a particular line length. Anything in CM, such as Amazing Grace, can be sung to the tune of anything else in CM, such as O Little Town of Bethlehem, or House of the Rising Sun, or Yellow Rose, or many more.