According to one report I found on Google the Christmas "carol" sung in The Great Escape by a POW choir in order to mask the noise of the tunnel workers was: "Frosty the Snowman" ...this is more like a Christmas song rather than a carol.
I think you're doing the same quiz as me. I downloaded the film and am still confused. They sing two different carols in the film but at neither time are they tunneling. The carols are "O Come All Ye Faithful" and "The Twelve Days of Christmas". I skipped through the film in 30 second intervals to find this out. I'll watch the film properly over the next week and will post back but I'm 99% certain those are the only two carols.