Waste into space is a bad idea, mostly because it is much easier to dispose of here on Earth than it is to put it up into space.
(Personally, I find the arguments about how hard waste is to dispose of deeply unconvincing. It's treated as a stick to beat people with, rather than an engineering problem to be solved, and one that is, frankly, almost entirely solved already in a number of ways.)
There are a wide variety of ways to use nuclear power for propulsion, such as: https://en.wikipedia.org/wiki/Nuclear_thermal_rocket NASA has recently been renewing investigations into this, as nuclear rockets have a much better story for getting into the rest of the solar system in some reasonably period of time.
You can ride on explosions, but, I mean, it's kinda brute force: https://en.wikipedia.org/wiki/Project_Orion_(nuclear_propuls... That said, if there was a "asteroid impact in three years that will wipe out humanity", this would be our best chance of lofting enough $STUFF into space to maybe survive as a species.
