Next I would look at wheel bearings, check for radial play at the rim in both horizontal and vertical planes. If this is the case you will need to remove the split pin, remove the castleated nut, and tighten this up , then the game becomes trying to get the split pin to line up. with the drillings in the stub axle, usually it involves filing the back of the nut.
Note there should be a tiny amount of play in cold bearings as they tighten up when used. Procedure for setting is do up nut to no more than tight with your hand 1/2 way along the spanner. Then loosen until movement can just be felt.
It's not widely realised but there should be two holes in the stub axle at 90 degrees to each other. The second one is usually blocked with grease and can be hard to find the first time it's used. Very rare that one of the two holes along with castelations in the nut don't give you the right setting.
But, as said by others, inspect with an assistant first as you do want to fix the right thing, not mess with the others.
