Carols for Choirs (OUP)has two tunes for 'Away in a Manger'; one by W.J. Kirkpatrick (1838-1920), which is the most commonly used, I think, and one quoted as a Traditional Normandy tune, with no date. I also sing it to an Traditional Germany melody, again undated.
I think the English trad. tune for 'O Liitle Town of Bethlehem', arr. by Vaughan Williams, is the most commonly used, the alternative I know is credited to Lewis H. Redner, undated.
'While Shepherds watched their flocks by Night' can be sung to a tune by Handel.