If the OD of the backplate shows 0.003" runout, the three jaw chuck will show at least as much when fitted to the backplate.
How much deflection you measure on a bar gripped in the chuck will depend on the diameter of the bar, and how hard you push or pull.
Obviously, if you push hard on a small diameter bar it will bend, apart from any bell mouthing in the chuck jaws.
The important figure for runout is that for material, measured close to the chuck. This may be a combination of Bearing clearance (may be different in one plane from another! ) Runout of backplate OD (Which is the register that locates the chuck ) Check that both are absolutely clean before reassembling. – Might be worth giving the backplate a very light skim on the face to JUST clean it up, with a small chamfer on the edge Wear between jaws and chuck body, differences between one jaw and anothe, and any bell mouthing of the jaws.
To improve matters you have to eliminate, as much as possible each error, until runout of a round bar in the chuck is minimal. It might be worth checking that the bar OD in several places, on the same line to be certain that it is circular.
There have been cases where steel has bee centreless ground is lobular rather than absolutely ground.s circular.
This lathe is old enough to be Imperial, so I would measure and work in Imperial units. Using Metric leadscrews will involve you in calculations at almost every turn (No pun! )
A lathe with 3 mm pitch leadscrews, so there are 118 thous for every complete turn of the handwheel..
A M12 leadscrew is 1.75 mm pitch, so each complete revolution will be 0.0689"
If you work in Metric, 70 divisions will make each one 0.025 mm, which is 0.000984", so just under a thou., which may get you close to the best of both worlds. ASSUMING that commercial studding is accurate and consistent in pitch.
With commercial studding and nuts, you will have to live with whatever backlash results.
Howard
Edited By Howard Lewis on 02/08/2021 08:34:22