What thread is the rear wheel axle nut?
Your rear hub would have been manufactured by someone other than Schwinn. Given the age it was probably manufactured by Sturmey-Archer or Sachs (now SRAM). Those links contain links to small parts for both hub brands.

Sheldon writes:


  The axle threading of most SRAM internal-gear hubs is unusual, 10.5mm x 26 TPI, but close enough to that of many Sturmey-Archer hubs (13/32 x 26 TPI) that Sturmey-Archer axle nuts are usable in a pinch.