Mark - I've got a '57 upright/spindle laying in my engineering lab that I use as "training device" for rookie mechanical engineers (I make them reverse engineer it).Both the upper and lower ball joint tapers are 1-1/2 inches per foot (the old school method of describing taper) - which translates to 7.153°. An experienced machinist will know which reaming tool is required.
The opening at the "big" end of the lower taper is .875 diameter. This allows the ball joint "pin" to protrude the correct amount for a full slotted nut with cotter.
The opening at the "big" end of the upper (small pin size) taper is .688 diameter.