Looking at the sizes of the shank of 9/16" nuts the OD of the shank appears in the range of 0.543" to 0.547" with 9/16" being 0.5625" so there is a fair tolerance.
The 5/8" shank nut should be OK to machine to size without compromising it, leaving a wall thickness no less than the 9/16" version, as long as the flat washer does not foul, and seats on the wheel, and a good radius is achieved where the turned down shank meets the hex.
From your pictures the wheels look "used" and as such I would ream the stud holes out, all to the same diameter, to give a nice bore and then machine the new nuts to suit working the size out from the tolerance stated above.
The only problem I can foresee that if the nuts were zinc coated this will be removed with the machining, so they may corrode, but if stainless this will not be a problem.