Basically:
Buy the bits!
Take off the new type stake nut and big washer behind it, take out the outer bearing, and remove the spacer behind it, discard or keep for napkin ring etc.
Bearing back in, fit lockwashers and double nuts to the old spec, tighten to the correct amount of play in the bearing, tighten the outer locknut and hammer the washer over to lock them. I think there is a post in the Tech Archive covering adjusting the old sort of wheel bearings so look at that if unsure.
Don't think I have forgotten anything
You will not remove any play in the bearing by "nipping up" the new type stake nuts (or even by giving it a #kin good heave on a 3 foot bar) because all you are doing is tightening the bearing against the solid steel spacer ring between the bearings, which will not make it get any closer to the other bearing unless you collapse the spacer ring which is not likely