The biggest problem with the cannon idea is that things traveling at that speed in the atmosphere tend to burn up before getting anywhere. And you get to start in the hardest part of the atmosphere to move in.
Unless you put the cannon on the moon. It's much easier to rocket to the moon than the sun. Now the railgun or whatever launcher doesn't have an atmosphere to deal with. Use the cannon to take away earth orbital velocity from the waste.
Throw in reusable rockets and... it's still really dumb. We have reactors that can make use of "spent" fuel - they're just not legal.
Throw in reusable rockets and... it's still really dumb. We have reactors that can make use of "spent" fuel - they're just not legal.