I can think of a few reasons this might be risky or impossible:
1. The risk of launching nuclear waste is too high, given the possibility of a failed launch spreading the waste in our atmosphere.
2. Even with an order of magnitude decrease in launch prices, it is still prohibitively expensive.
3. For engineering reasons, it's nearly impossible to launch a rocket at the sun.
If nuclear rockets or reactors are built in space, would this be a possible way of handling nuclear waste from reactors already in space?
The short answer is: Counterintuitively it would take less energy to launch a spacecraft to another star than our own sun
As for why not into the space in general:
Risk of rockets failing. Lancing probes with small nuclear batteries is controversial. Think about launching tens of tons of nuclear waste. Almost 100,000 tonnes of nuclear waste is produced per year. That's clearly too much to launch.
If we would use SpaceX Falcon Heavy to launch only the highly radioactive waste )12,000 tons per year) for $2,500 per kg. It would cost $30 billion per year and 240 Falcon Heavy launches per year. Some of them will fall down, explode etc. and that must be dealt with.
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https://www.popularmechanics.com/space/rockets/a21896/why-we...
https://www.forbes.com/sites/startswithabang/2019/09/20/this...
https://www.universetoday.com/133317/can-we-launch-nuclear-w...
https://astronomy.com/news/2016/07/heres-why-we-cant-just-ro...
The math might change once we get a space elevator and a solar sail since it would be cheaper to get things into orbit and then parachute them into the sun.
Another idea would be to bury it into a really deep hole (https://en.wikipedia.org/wiki/Deep_borehole_disposal) or a part of the tectonic plates that would quickly take it into the Earth's core (https://en.wikipedia.org/wiki/Ocean_floor_disposal).
Source: https://www.theatlantic.com/science/archive/2018/08/parker-s...
> If nuclear rockets or reactors are built in space, would this be a possible way of handling nuclear waste from reactors already in space?
If you're not already falling towards the Sun, it must mean you're in some sort of orbit around it. So you'd still need to cancel that out.
2 and 3 are almost the same thing with “difficult” and “expensive” being closely related.