I remember it from BBC radio too - stentorian voice with a rather clunky piano accompaniment.

There are only seven hits in Roud (one of which is a different song), and it looks like Mason is the source for both Broadwood / Maitland and Ritchie, i.e. the sole example of the song in tradition. Maybe a one-off parody as SG suggests - but though the tune is related to that of 'I Saw Three Ships', there are several differences, not least the recasting in the minor.