Hi, it will be a bit of a bugger if the straight part of the spindle bore that locates the back end of the R8 collet is running out......once that has been ground to size it's there for good......you cannot regrind it a few tenths bigger to cure runout or the collet will end wobble and run out.

But you can touch up the front taper with a die grinder a few tenths and if you have access to a lathe or someone with one then you're home and dry.

With a lathe you would hold the one end in the chuck and run the spindle on its front bearing diam in a fixed steady and use a die grinder....whatever...... with the compound slide set over to the taper angle.

Without any access to any machinery at all or any ability to build a pseudo lathe.....that is, a set of linear rails and a pair of plumber block bearings for the headstock etc etc...…..anything you do to the spindle will probably ruin it for any further use despite the slight runout, so I would advise leaving it as is and pretend you didn't know it ran out...….or buy another spindle as a spare part.
Ian.