According to the print I have, the tang screw is 12-28.  1.59" length under 90 degree countersunk type oval head.  1.71" OAL.  .069" countersink taper length.  .040" straight section .343" diameter between countersink and oval head. .406" thread length.  .205" shank and major thread diameter.

Vernier tang sight requires additional length and possibly different head style.

I didn't tear a rifle down to confirm these dimensions, so somebody please correct me if they are wrong. 

Hayface

Frank,

from an old Marbles chart for the WRA short tang sight mounting screw:
Whitworth  0.185" x 35 1/2

The Prints?? that exist call for a .185-35 1/2 or 35.5 pitch as someone else has pointed out. Now in the older world we all came from the gearing to cut 35.5 is not easily available. Winchester probably spent a lot of time to arrange that or had some type of geometric dies made. In today's world of CNC machines it is just programed number 35.5 comes out to .02816. Now a 36 pitch comes out to .02777 per turn. So the difference is .02816 Minus .02777 = .00039 per turn. Now if you cut a 10-36 pitch and it only for 3 to 5 turns and you cut it just a little loose you will be OK as long as it does not bother you. I work in the medical field making operating instruments and everything is Metric. We have to hold it. We ended up buying a Programming drive that I can program lead screw feed rate for and pitch desired Metric or Decimal. Pain in the A$$ sometimes but it works. So have many Gunsmith and Hobby mechanics over the years using 10-36 for Highwalls just to keep them going. HTH, Regards, FITZ.
PS the total accumulated error for 4 turns at a 36 pitch comes 
out to minus .00156 lead error. FITZ. 

Frank-
I would confer with the fine folks at MVA sights
Chuck V

I use 10x36, have the tap and die. It's so close that over the short length of the screw it does no make much difference.