In post #575, the indicator tip was on the chuck body. What is the runout when on the back plate? I would check the outer edge just to the left of the chuck body, then check back side of the back plate, (only if it has been machined).
A 0.001+/- at the spindle shoulder runout will be larger farther away from the shoulder.
If there was a bit more material between the front of the shoulder and head stock, I would be tempted to take a very light facing cut off the shoulder.
There is a possibility that the collet chuck body was turned end for end when machining spindle mounting thread and then the taper and thread for collet and nut. Result, possible non parallel machining.
In picture post #500, were the bronze stock pieces with the lathe/spindle, as thought might have been to make bushings for the other spindle.