Yeah, it seems that the sun just isn't big enough for that sort of thing. As the universe is also expanding, the uranium we find in this system is probably locked in here until the heat death of the universe.

If you could expect the elemental cycle to continue forever, you could argue that the elements of nuclear waste would eventually end up in a star which would recycle them into new uranium. The expanding universe makes that highly unlikely, so even in extremely long time scales, nuclear power isn't really renewable at all.

If the big crunch hypothesis turns out to be true, there would be an infinite cycle of new universes, and that would make every element renewable in extremely long time scales. However, the ultimate fate of the universe isn't particularly clear at the moment, so who knows really. For the time being, I'll just assume that the heat death is the most likely outcome.