I took it apart again and checked the shaft and found it was good and true. So I re-assembled and it ran smoothly. I must have bodged something first time!

Drill back together and it runs quieter and less wobbly now. Sadly still a little bit of runout which I think is from the chuck arbour. Good enough for now and I'm not sure how to remove the chuck as the morse taper is directly on the end of the spindle, meaning I have no way to press the chuck away from the spindle shaft. I don't think it was meant to be removed. It's only a cheap Axminster WD13S (about £90 I think).

Tempted by a Bosh PBD40. Any reviews on this?