To be on one’s Jack Jones is to be alone; on one’s own.
This imperfect piece of cockney rhyming slang appears to derive from the music hall song ’E Dunno Where ’E Are, written in the early 1890s by Fred Eplett and made famous by Gus Elen. The lyric concerns one Jack Jones, a former Covent Garden market porter who has come into some money and now considers himself above his old workmates. He reads the Telegraph instead of the Star, calls his mother ‘ma’ instead of ‘muvver’ and stands alone at the bar drinking Scotch and soda.
However, modern usage of the term rarely implies aloofness on the part of the person alone; more often it is close to the opposite – he or she may feel abandoned. For example, “You lot went off and left me on my Jack Jones!” [‘my’ is invariably pronounced ‘me’ in this context].
And why you all over there on your Jack Jones?
You need to let me get behind your backbone.
D Mills and C Harris: ‘Dance wiv Me’ (song by Dizzee Rascal feat. Calvin Harris and Chrome) (2008)