Howard,
It looks like a standard Boxford tailstock, if so the parts you need are easily made.
1. a 2¼" length of 5/16" steel rod (round bar), threaded 5/16" BSW at each end, one end threaded for 5/8", the other for 7/8". The shorter thread screws into the bottom of the hole in the casting.
2. a 3/4" length of 5/8" dia phosphor bronze or brass, drilled through clearance for the threaded rod, with a corner cut off.
3. a ball handle tapped 5/16" BSW, or a nut and spanner.
see below. The rod is screwed into the base of the hole in the casting, the PB/Brass piece (sliding cotter) slides over it with the cutaway down and inwards and the handle/nut presses it down. There is a "window" inside the casting into which the barrel protrudes and the sliding cotter presses on the barrel through it. There is only one sliding cotter, the form for the second cotter is machined inside the casting