Put Dykem (bluing) in the shaft and check where the nut is touching. If the peak of the thread is sharp what feels like a good fit might be only the peak touching or possibly the thread angle is wrong and touching the corner of the V. The peak is commonly either flat or rounded to height/8 on National threads and height/6 on Whitworth
The triangular height for a 20 TPI thread (0.050" pitch) are:
National - 0.86603 * pitch = 0.043315"Whitworth
- 0.960491 * pitch = 0.048025"
Subtract half that from the basic major diameter to get the pitch diameter.
National - 0.60335"Whitworth
The theoretical O.D. is
National 0.625 - 0.043315 / 8 * 2 = 0.61417"
Whiworth 0.625 - 0.048025 / 6 * 2 = 0.60899"
Was the shaft cut oversize or to a metric dimension? 20 TPI is 1.27mm. If the thread is actually metric a 20 TPI nut will fit if the depth is cut far enough and the length of engagement is short enough.