This reminds me of Edward T. Cone's idea of "beyond analysis." Some things, Cone says, are just "beyond analysis." Why did the composer choose this key, with this time signature? Why does the oboe start and not the clarinet? Why begin with an ascending third instead of a descending third? These are examples of questions that are "beyond analysis"; they're that way because the composer said so, end of story, and there's no point in searching for rationale.
With that said, I doubt there will be any definitive answer unless we hear something from the original composer, but the text is one possibility.
On the twelfth day of Christmas
My true love came to me.
Note that the first line has 7 syllables; although one could certainly squeeze that into a measure of 3/4, a carol for all to sing will likely use rhythms more akin to natural speech. As such, this couplet naturally lends itself to 4/4, in my opinion. (As does the final line, "and a partridge in a pear tree," which has 8 syllables.)
Meanwhile, the gifts themselves are all lines of 5 syllables, with some of them ("three French hens") only three; it's tough to stretch these out into a measure of 4/4 with a rhythm similar to natural speech, so this couplet naturally lends itself to 3/4. (Again, in my opinion.)
As for the five gold(en) rings: I've always felt that bit was intentionally overly dramatic to give the carol a little bit of humor. And this is because the music is stretched out (and comically so) to fill two bars of 4/4 instead of just one bar of the preceding 3/4.
But again, I say all of this with the caveat that this is just my reading of the piece.