The volume of nuclear waste that must be stored is vastly smaller (by orders of magnitude) than the amount of emissions (and often ashes) from fossil fuels. About the same size as the radioactive tailings from mining rare earths for wind turbines (depending on the design).

And a big difference is that nuclear waste IS stored. In engineered containers. With the price paid by the nuclear power provider. Unlike CO2 emissions from fossil fuel power plants.

So I'd say, given the orders of magnitude difference, it is pretty much the full story. Except to note the waste issue for nuclear is vastly exaggerated.

Said waste does not contribute to global warming/climate change in bad ways.