Silly question but are you tightening these two nuts together or separately, after you've tightened the bearings up?
Separately. I tightened the lower 1st, then the upper onto the lower to (attempt to) hold in place.
All very good suggestions in this thread - I will investigate this evening.
"Proper" technique for a locknut pair is to use two spanners, one to hold the nut against the bearing still, and tighten the locknut down with the other, pretty sure it's why cone spanners used to come in pairs (or threes in some cases, ancient shimano stuff springs to mind, not sure why)
I have a vague recollection that my first headset spanner came as one of a pair. One end of each was sized for a headset, the other ends were respectively a peg spanner and a lock ring spanner for the adjustable end of bottom brackets. (think the headset end of the spanner was used for the right hand cup, but it was 25 years ago, so not sure