Have you tried to get the younger William's death certificate? Hopefully it has who his mother was. His obitaury might help too especially if it lists his siblings. As a Civil War veteran who died well after the war I would imagine they had a nice write up on him.
Have you gotten a copy of his pension from the National Archives? If so I would be surprised if it didn't say where he was born. Have you also gotten the service from, also from the National Archives, for both father and son? They generally contain little info but even one clue can be helpful.
The elder William's widow might have applied for a pension. Sometimes the children applied for a pension.
The only two entries I found for Fords is that of a Henry Ford who enlisted as a private in Company A on April 21, 1861 and was mustered out July 31, 1861. A William Ford enlisted as a private in Company C on Sept 1, 1861 and mustered out Sept 8, 1864. I could not find a Henry or William Ford who died while a member of this regiment. I could find nothing about either being a musician. This regiment was formed in Philadelphia. It doesn't mean everyone was from Philadelphia but most were.