The register diameter you can measure, and the depth will be measurable too, less an allowance for the threading runout groove. You will need to turn to 1.250 +.00002/ +.00005 if you want it to really run true. (Keep it well oiled with 3:1) As long as it clears the actual depth of the register is not critical.
The ones I have cut for mounting stuff on a VDH, and putting a boring head into the Dore Westbury have had registers about 1/2″ long. The one supplied on the BS0 dividing head, which is perfectly secure is not much more than 3/8″ long.
The thread is 1 1/8 x 12 55deg Whit form, but you wil need to ensure the crests are truncated a fraction, to be sure it doesn’t hang and the register and back flange are doing the aligning..