Here's that song, and a bit more from the beginning of the film.
It's that as good a film as Scrooge is, Bricusse's research staff was lousy. Because 14 years before, in 1956, there was a live TV musical version of A Christmas Carol that was called The Stingiest Man in Town and starred the legendary film actor, best known as playing Sherlock Holmes, Basil Rathbone as Scrooge. (A live TV musical for the holiday, who'd have thought...?!) And the very first song in the show was called -- yes, you guessed it -- "A Christmas Carol."
The music for the show was written by Fred Spielman, with lyrics by Janice Torre. It's not remotely distinguished or memorable, but has quite a few very nice things in it. The production was done for the series, The Alcoa Hour, and also starred Martyn Green; a mix of opera singers, such as Patrice Munsel and Robert Weede (who starred in Frank Loesser's The Most Happy Fella) and pop stars of the day, including Johnny Desmond and The Four Lads, as sort of traveling minstrels.
So, continuing our holiday theme of unknown Christmas songs from musicals, here is the earlier song, "A Christmas Carol," sung by The Four Lads -- leading into "An Old-Fashioned Christmas" (sung by Vic Damone), from The Stingiest Man in Town. That the researchers couldn't find. But we think you fine folks deserve better...