My son brought this home from school last week.

The song appears in this 1969 book:
Sally Go Round the Sun

And at the end of the book, it references 'Songs and Slang of the British Soldier 1914-1918' by John Brophy and Eric Partridge, 1930.

But I have had no luck tracking down a digital copy of that book. There was another song called 'Down Where the Watermelons Grow', which is possibly confused with this one.

In any case, my son had fun making this recording today.

It is a fabulous campfire/kids song and I am amazed that I never heard it growing up in Australia in the 1980s.