It’s alll in the way you set the bearings up then. If you have no manual with torque specs etc, best to adjust the spindle nuts so the bearings have zero clearance and a very slight preload.
Measure for clearance by holding a 300mm length of bar or pipe in the chuck and yank the end of the bar up and down while measuring spindle movement right next to the headstock with a dial indicator. Indicator base should be on the bed not the wobbly carriage etc and the indicator plunger should rest directly on the lathe spindle between the chuck and the headstock somewhere.
It should have zero measureable movement. Then tighten up the spindle nuts just a bit so there is slight resistance to rotating by hand, ie a small preload on the bearings. You should be able to turn the spindles by lightly gripping with fingers, not needing to grab it like an axe handle.
Once you have done that you can then check spindle runout with the dial indicator touching directly on the spindle in that same area between the chuck and headstock somewhere. Or even better, on the spindle register where the chuck locates.