(You can make those parts inert and safe to handle. Even the fuel rods of current reactors are safe to touch before they are spent.)
I just don't see how we can get there from here. Nuclear isn't particularly cheap (but yes, a good chunk of the expense comes from regulation). So there's no obvious reason to choose that over any alternative. We could certainly change tax structure to force oil and coal to be more expensive, but even simple changes to account for externalities are met with massive massive resistance.
But yes, we won't see that..
I wonder why noone has ever tried selling eg a carbon tax (or any kind of broad and useful tax) in combination with basic income? If you earmark the windfall from a carbon tax to be equally shared by all residents, you get quite a few natural allies and take the wind out of the sail of arguments along the lines of "think of all the poor people".
What? Fuel rods consist of uranium, which alpha-decays into thorium (a sheet of paper is enough to shield the alpha particles), but the next decay step is beta-decay to thorium, and that needs more shielding. No way one can touch a fuel rod without getting a load of radiation.
There's also an interesting anecdote about spent fuel here [2]. You can get pretty close, if it's under water... but it gets very dangerous very fast.
[1] https://en.wikipedia.org/wiki/Spent_fuel_pool [2] https://what-if.xkcd.com/29/