Although the name has the same origin, it might help to know that it's pronounced differently:
/no-ell/ - a Christmas carol, Christmas
/no-'l/ - the man's first name
That is, in Noel Coward's name, the accent is on the first syllable, so I don't think it's usually written with a dieresis in English. I think the same is probably true for Byron, but I would have to look that up and I'm not where my books are.
Though the dieresis with the name in German might help keep people from misreading it as 'Nöl.' (-;
Back to the regular word: The two meanings from the dictionary would be (1) Weihnachtslied, and (2) Weihnachten. But you can't always use those translations in Christmas carols, which are now the only context where the word 'noel' is often found. There the meaning is often more like 'glad tidings' or 'good news.'
In 'The First Noel,' the angels 'say' the noel rather than sing it (probably for the sake of the rhyme), but the 'first noel' is basically the Gloria in excelsis, and we often think of the angels as singing it. So I think the best translation to help people understand the words might be something like 'Weihnachtsbotschaft, -kunde, -nachricht.'
The first noel the angels did say
Die erste Weihnachtsbotschaft, die die Engel sagten (= sangen),
was to certain poor shepherds in fields as they lay.
galt gewissen armen Hirten, als sie in den Feldern liegten.
In English Christmas carols, 'noel' in refrains is often just a repeated sound, like 'Fa la la.' It means something like 'Sing the good news of Christmas,' but in only two syllables *g*, so I don't think it's very translatable.
In a context like 'Noel, noel,' the closest equivalent in German might be something like 'Eia, eia.' I'm not sure what that actually means, if anything, but I'm sure someone here knows. (-: