Well, you separate the spent fuel into chemical components by suitable chemical processes. One of those components is plutonium salts. So, yes, reprocessing is a rather direct way to nuke materials.
There are reprocessing schemes like pyroprocessing that are less useful for producing bomb materials. But yes, currently at industrial scale the method of choice is PUREX, which AFAIK is the same process which is used for producing Pu for weapons. Although when reprocessing high burnup spent fuel from commercial reactors you get Pu with an isotopic composition poorly suited for weapons.
