I strongly suspect that the Song Spinners (NZ not Liverpool) including Arthur Weller and Neil Colquhoun, looking for some whaling songs suitable for school children, found a selection of tall tales and songs - Richard Chase's book and Burl Ives songs were available in NZ in the 1950s, and they include silly songs that kids love. If the group functioned as a 'workshop of song', each contributing ideas for the arrangement and presentation of the songs, as the record states, I can imagine them finding the NZ Weller family story and then looking for a song to adapt and turn into a 'NZ Folksong". They never recorded it - maybe they didn't finish their song and it took Colquhoun a few years more to work it out to an acceptable level. Maybe others helped him in the 1960s, after the group disbanded.
The New Zealand Folklore Society never found it in 1969, when we sent Phil Garland on a hunt for the 'great NZ Folksong' and he travelled all around the South Island looking for songs. Mostly he found poems.
Robyn