Energy density isn't, alone, determinant here.
The most determinant question is: after taking into account all measures aiming at reducing renewables' production 'intermittency' ('variability' is more adequate), which are: spreading a mix (wind, solar, geothermic...) of renewable sources on a continent, enhancing grid interconnections among nations for them so support each other (objective already very actively and more and more pursued in Europe), storing at continental-level renewables' over-productions (thanks to hydro, hydrogen...) and using it on a clean 'backup' compensating for renewables' 'intermittency', demand response... ... then how much energy do we need to store in order to compensate for any remaining 'intermittency'?
Given that the average electric car battery can power the average home for quite a while (days), and that other EV batteries will be online...
Moreover each year 7% to 10% of gridpower is in France (fully nuclearized) produced thanks to fossil fuel, and its electricity is low-emitting, giving a quite comfy last-resort.
> And how much mining is required obtain those materials
Those materials are recyclable and have substitutes.
Uranium isn't (in practice) recyclable and doesn't have any substitute: this is eternal mining of an ever-rarefied source.
> shipping them around the world in diesel powered boats?
This is a valid point, enforcing the need for each continent hosting cooperating nations to produce its own production units.
On the other hand uranium reserves at current conditions can provide for at best 200 years (more probably 130 years: https://en.wikipedia.org/wiki/Peak_uranium ), therefore the conditions (price and emissions, as the raise as the ore grade lowers may quickly worsen if some 'nuclear renaissance' stems reactors building projects. Who will take the risk and invest?
Regular houses are too small to make it possible to install enough solar to charge the car though.
That's without even considering appartment buildings.
On average the car autonomy already is way above the daily commute, and this margin is quickly growing.
There are more and more parking places (private and public) connected to the grid.
A bit of intelligence ( https://en.wikipedia.org/wiki/Smart_grid ) will orchestrate.
This would only work if the added charge on night-time electricity also accounts for battery cycling cost which I doubt it would.
https://en.wikipedia.org/wiki/Negative_pricing#Electricity
Reciprocally as more and more less tolerated grid backup (greenhouse-gas emitting fossils or dangerous nuclear) disappear episodes of under-production will be more common, and during those electricity price will soar.
No level of organization is sufficient: https://en.wikipedia.org/wiki/Forsmark_Nuclear_Power_Plant#J...
- there was no one really in charge of such matters during most of the pertinent decades (first leg of Mao's regime): politics were the sole concern, leading to well-known disasters such as the Great Leap Forward ( https://en.wikipedia.org/wiki/Great_Leap_Forward ) and disdain (or worse) towards whistle-blowers https://en.wikipedia.org/wiki/Banqiao_Dam#Whistle-blower )
- the (Soviet) designers had no skin in the game
A nuclear reactor built and used in such a context may trigger some bad events in way less than 23 years, as a mere few seconds are sufficient for it to do so.
The amount of damage/victims linked to nuclear is a matter of debate, and a final count will only be possible after its very last hot waste will be cold.
Case in point (Chernobyl):
https://www.smithsonianmag.com/science-nature/forests-around...
https://knowablemagazine.org/content/article/food-environmen...
https://en.wikipedia.org/wiki/Chernobyl:_Consequences_of_the...
Energy storage: vehicles batteries are a game changer ( https://en.wikipedia.org/wiki/Vehicle-to-grid ). Grid backup will also use existing hydro along with turbo-alternators burning green-hydrogen ( https://en.wikipedia.org/wiki/Hydrogen_economy#Energy_system... ).