His reason for wanting to pull the spindle is to break loose and remove a stuck chuck, not necessarily to change the bearings.
REF:
Atlas Lathe 06 3950 MK2 CI Ball Rev 1.pdf file in Downloads, page 15.
Mathewsx, I've never actually had my hands on a MK2 but my assumption from looking at the exploded view parts list is that you would first remove everything on the spindle external to the left side of the headstock. From your description you would next loosen the collar set screw and remove the 123-140 threaded collar. Then remove the 341-374 Large Spindle Back Gear and two #404 Woodruff Keys. One of the keys may already be removed. Then from the drawing, it looks like the spindle will go out the right side of the headstock, leaving the left hand bearing in the left end of the headstock and maybe the right hand one, although it bay come out on the spindle.
Anyway, with the spindle out of the headstock, you should be able to get a strap wrench around it near the right end. Put a large piece of hex stock in the chuck with the jaws solidly tightened on it. If you have one, you can now use an impact tool (either air or electric) and socket on the hex stock to break loose the chuck.