Nuclear fuel is extremely difficult to refine into weapons grade material - even in the best of circumstances.
Conventional weapons material is either super enriched uranium (virtually pure U235) or highly isotopically pure plutonium (virtually pure Pu239, with definitely no Pu238 and virtually no Pu240).
Weapons material can be obtained either from super-high performance uranium enrichment equipment, or from 'breeder' reactors operated in a specific way.
During operation of a conventional reactor, as the U235 burns it irradiates the U238 converting some of it into Pu239. In military Pu reactors, the fuel is loaded for a few days, then removed and the plutonium extracted.
In commercial reactors where the fuel is left in place for up to 2 years, a significant amount of the Pu239 formed is converted into Pu240. This degrades the quality of the plutonium and makes it unsuitable for weapons.
Unlike uranium, where enrichment technology has existed for 50 years, there is currently no practical way to enrich plutonium. It has to be produced in the isotopic purity required.
Spent nuclear fuel contains a significant amount of unburned U235 - to the point, where the spent fuel is still significantly enriched. This could be extracted, and re-enriched. But as you still need enrichment facilities, it would be better to use natural uranium to avoid dealing with radioactive materials.
A new nuclear reactor design called the pebble bed reactor, is supposed to increase security still further. The fuel consists of sugar grain sized specks of uranium coated with carbon, silicon and other materials and embedded in large solid carbon 'pebbles' the size of tennis balls. There is currently no practical way to extract the uranium/plutonium from these pebbles in reasonable purity - as they will always be contaminated by the carbon/silicon. The problem with this design commercially, is that because the uranium could not be extracted from the waste, it wastes uranium, and needs a huge amount of space in long-term storage.