Nuclear rockets won't "ignite" until they're in a nuclear safe orbit, and it will be on a escape trajectory away from Earth. And a large rocket as you propose likely won't be able to have any prolonged operation in LEO, it would be above where the majority of debris is.

All of this ignores the simple fact that this isn't a video game, you don't just drive around picking up trash. You can use it to send trash away once you collect it, but it's not even remotely a good proposal on how to collect it.