Sometimes it is difficult to attribute the carol. In a number of cases, a German (or whatever) rhyme has been taken by an Englishman and rewritten to fit a French tune, and in turn this piece has been re-arranged in both words and music by an American organist, etc. etc.

Poppagator, here is what Hymns and Carols of Christmas says about "Away in a(the) Manger." Although some believe that this carol was penned by Martin Luther,...... it is almost certainly of late 19th century American origin." Two tunes are mentioned, both seemingly American, but one preferred in England- Away in a manger
The main line of evidence is that it does not appear anywhere before late 19th c.