Waste disposal isn't that hard. The Onkalo nuclear waste repository in Finland pretty much has it covered. First, you find a hard-rock mountain where geology shows nothing much happening in the last few million years, and there's nothing worth mining. Worldwide, there are many mountain ranges like that. Then tunnel way down, but preferably above the water table. Drill holes in the tunnel floor. Put waste in suitable…
How do we communicate to civilisations in the future what we did in that mountain?
Please, let us know your plan.
P.S. This is not a hypothetical problem, like the demands you are making of nuclear. This is a current and ongoing one. Your green energy isn't as green as you claim it is, please tell us your solution.