Threads are always under the nominal size so a 5/8" nut is never going onto a 16mm threaded shaft even with the same TPI. What TPI is the shaft, if it is 20 TPI then run a 5/8" 20TPI die over it or get some one with a lathe that can be set to 20 TPI to cut the thread to the proper OD. If its not 20 TPI then that old odd nut looks very attractive and you can make the circular section a hex.