I remember in Stevenage UK in the late 50's Tonibell (who were originally Toni's round our way) used to have their own tune (it even had words.

Tonibell Tonibell Tonibell time
Tonibell Tonibell Tonibell time
Friendly guy, come and buy when the bells chime
He comes your way at, everyday at, Tonibell time.

And Walls Icecream also had their own chime which was just 5 notes
A,D,F#.A.D (The last D being a high D)

Don't remember Lyons or Neilson's though