In a 75 year old British machine the dimensions and threads will be Imperial (BSW and BSF, possibly with BA for gib strips)
The spindle run out may improve as you adjust the bearings.
Since the chuck is also old, a three jaw will not hold anything concentric, unless you are fabulously lucky (A good 3 jaw might hold work concentric to about 0.003″. An old worn one might be 0.030″ or more!)
As said, even with such eccentricities, once work has been turned down, the reduced diameter will be concentric. If you need to have two or more diameters concentric, turn then all at one time, without disturbing anything.
If you want to start with absolute concentricity, you will need a 4 jaw independent chuck a DTI and magnetic base.
Fitting a 4 jaw, if you don’t have one, might involve some machining to produce a backplate which is compatible with the spindle nose.
This might entail some internal screwcutting, as well and turning the OD to make the register, then facing it. So will get [practice on rotating handwheels at a slow steady rate to get a fine feed.
All good practice to become familiar with the lathe.
Howard