I forgot to respond to your comment about the Monkees song, but basically I think you're right, that it's something that came from them rather than a producer. Back in those days everybody recorded the same songs (I've seen enough of my aunts records from that era to know that once a song became a hit, all and sundry would do their own version.) And I don't recall seeing or hearing this particular Christmas song on any of my aunts old albums.

It's a truly lovely song btw.