[DennisP]

Reprocessing just reduces the amount of waste. As far as I know, France is not doing the next step, "use modern reactor designs that actually clean up old fuel from light water reactors."

Conventional nuclear reactors are limited in "burnup," the amount of fissile fuel they manage to fission before neutron poisons shut down the reaction. Reprocessing removes the neutron poisons so you can fission more of the fuel. (Some of the waste products of fission are neutron poisons, meaning they soak up neutrons so they can't trigger fission reactions.)

Even after reprocessing, the bulk of the long-term radioactivity comes from plutonium and other transuranics (elements heavier than uranium). Those are made when uranium (or heavier) atoms absorb neutrons without fissioning. Reprocessing doesn't help remove those, but advanced reactors can fission those just fine. They can also fission U238, which is the bulk of nuclear waste (though not really radioactive by itself).

What's left after all that is just the fission products, which is about 1% of conventional reactor waste. Encase them in glass, bury them, and they'll be back to the radioactivity of the original ore in about 300 years.