Don’t trust that the axle had the correct nut.
There are six common axle diameters: 9mm and 10mm with 1mm thread, often hollow, and 5/16″ and 3/8″ with 24 tpi or 26 tpi. And a caliper won’t always help: 24 and 26 tpi fall either side of 1mm, and 3/8″ falls between 9mm and 10mm.
A washer with the nut, or a washer integrated with the nut, not only lessens the nut chewing up the frame, but also limits the axle slipping as the nut is tightened.
Solid axles with nuts have some nice qualities besides being stronger than hollow axles with quick releases. You can get the wheels solidly in the bike without putting it on the ground. And it’s much easier to center a rear wheel in horizontal dropouts than with a quick release.
To get an axle with nuts solidly in a bike, start by holding the wheel with the axle firmly in the fork end or vertical dropout and tightening one nut, then push the top of the wheel toward that nut so the axle comes up firmly in the fork end or vertical dropout with the loose nut, then tighten that nut. Now go back and loosen the first nut, push the top of the wheel away from it so that the axle comes up firmly in the fork end or vertical dropout, and tighten the nut.
A similar procedure is used for singled speed or fixed cog rear wheels in bikes with horizontal dropouts or track ends.