Two socket fluorescent lamp OFF switch hot / not working with single bulb
A quick search of ballasts revealed electronic ballasts that support (2) F15T8.  

Go straight to that, don't waste any time with old magnetic ballasts that buzz and flicker on a good day.



An instant-start ballast does that by shocking the tubes lit, but this wears the tubes faster.  It typically needs 3 wires for a 2-tube setup, and they can run independently. 

A rapid-start or programmed-start ballast makes use of the preheat coils in the tubes (automatically doing what the start switch did), causing an easier start.  The rapid-start ballast is guessing.  Both of them need typically 6 wires for 2 tubes, so you'd need to add wires to the neck conduit.  They usually need both to be present/working.