The Larks of Dean Quire have a great deal of material to work with from the manuscripts of the original "Larks". This includes carols as well as the day to day hymns. Some of these are very similar to the ones in the Sheffield tradition and others not.
I would not call it a living tradition but we do perform concerts and include carols in the correct season!

One thing that I have noticed is that the "Mortals awake" set of lyrics do not seem to feature that strongly in the Sheffield tradition, but does in Lancashire.