What would happen, if you managed to make sure all this stuff got escape velocity, is you would wind-up with a bunch of junk (literally) in an orbit around the sun that would inherently pass through Earth's orbit - as long as it doesn't have solar escape velocity, once a thing is moving on pure momentum, it is in orbit and since the orbit fairly closely an ellipse so the object will return. As the article mentions, probably not when the earth return to the point. But if you keep throwing stuff "out there", the chances of a return are naturally going to increase.
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.
Given that the past years have "proven" that it's "perfectly safe" to set off hydrogen bombs in the desert; I think it'd be reasonable to assume that it's quite possible to launch a container of junk into space. Not sur why one would care if it hits the sun or not...
Iff one can make a container that can survive the blast/lift-off -- one wouldn't need to worry about the launch vehicle exploding in-atmosphere: there'd be no rocket fuel. Added bonus: find a way to use existing war heads, and combine with nuclear disarmament...
But that would carry the risk of the container filled with nuclear waste colliding with earth at some point in time. And considering there is over 200k tonnes of nuclear wastes, that risk would be a lot larger
A solar powered mass-driver could sling a steady stream of pellets from Earth orbit into Solar non-orbit using existing technology.
That being said, burying it deep near the the start of a plate subduction zone makes at least as much sense.