You don't actually need to mine more uranium though. You can run certain nuclear designs on Thorium, Plutonium from weapon stocks, or even waste from other reactors. Current generation nuclear designs are laughably inefficient at using the nuclear fuels we have available, and I fully understand why people don't support them.

Realistically though I don't ever expect nuclear fission to be as cheap as renewables in most areas. In some places nuclear or another power source is always going to be needed though just because renewables are not practical in certain conditions.

In the long term the answer is almost certainly going to be nuclear fusion or another future power source like neutrino voltaic. Solar and wind power are ultimately just offshoots of fusion, and so is fission if you think about where uranium, thorium and so on come from. In fact all power we know of seems to come from either gravity or some kind of nuclear reaction (inc. geothermal and fossil fuels).