It’s super easy actually. You start out by just exchanging all the money already in the banks. Then, as businesses and people deposit their old money into the banks, you swap that out too. It might take a bit to swap out every last dollar, but after a short time you would get the overwhelming majority. The only old bills left would be the wad of cash under grandmas bed.

Now you could speed the process up by saying on x date the old money is no good, but that’s considered a bad idea because it lowers faith in your currency