Rechambering a .223 Remington to .22-250 is simply NOT a job to do by hand. It's much too tedious to try and remove that much material, and you are just asking for a ruined chamber, reamer, or both.
You need a lathe with the proper accessories and technique to do it right. For one thing, the pilot of the .22-250 reamer is not long enough to enter the barrel past the .223 chamber and provide guidance. You will have to drill out the old chamber to reduce the amount of chamber to ream out. Headspace gages are also essential.
If you have never done this before, or don't have a machinist's background, it's almost certain that paying a qualified gunsmith to do it is a better alternative.