Well spotted, Splott Man. I don't know the answer, butâ€"

There's also JOHN WILLIE'S FERRET, sung by the Oldham Tinkers.

The Oldham tinkers also recorded a song called JOHN WILLIE'S HORSE, but it hasn't been posted at Mudcat.

There is a brand of beer called, officially, J.W. Lees from Manchester, but it is often referred to colloquially as John Willie Lees (or incorrectly as John Willie Lee's). It has been mentioned several times at Mudcat.

George Formby (Senior) recorded a song called JOHN WILLIE, COME ON in 1908. It also hasn't been posted at Mudcat.

Gracie Fields' song OUT IN THE COLD, COLD SNOW mentions a character called John Willie Joe.

In her song WHAT CAN YOU GIVE A NUDIST ON HIS BIRTHDAY? Gracie says that her brother, the nudist, is named John Willy (or John Willie).

In 2003, there was a Mudcatter who posted once (only once!) using the name John Willie.

In another Gracie Fields song, HERBERT HENRY HEPPLETHWAITE had a friend named John Willie Pratt.

Mudcatter Rob Naylor reported that he had a great-uncle named John Willie Priestley.

Mudcatter GRex reported that his father was called John Willie

Raggytash requested chords to JOHN WILLIE'S RAGTIME BAND, but it was never fulfilled. (And the lyrics weren't posted either).

A search in WorldCat.org reveals that there are songs called
FETCH JOHN WILLIE (1910)
HAVE YOU SEEN MY JOHN WILLIE (1914)
JOHN WILLY (1927)

After reviewing the data, here's my opinion: John and William are common enough Christian names, and knowing someone named John Willie is just a natural consequence of the custom, common in some regions, of denoting people by two Christian names.

This is an interesting enough topic, I think I will start a thread about it.