Nuclear power stations will have their chain reactions shut down:
However, there is a serious problem remaining, namely decay heat (https://en.wikipedia.org/wiki/Decay_heat)
Quotes from that article:
After one year, typical spent nuclear fuel generates about 10 kW of decay heat per tonne, decreasing to about 1 kW/t after ten years. Hence effective active or passive cooling for spent nuclear fuel is required for a number of years.
If no cooling system is working to remove the decay heat from a crippled and newly shut down reactor, the decay heat may cause the core of the reactor to reach unsafe temperatures within a few hours or days, depending upon the type of core. These extreme temperatures can lead to minor fuel damage (e.g. a few fuel particle failures (0.1 to 0.5%) in a graphite moderated gas-cooled design or even major core structural damage (partial meltdown) in a light water reactor or liquid metal fast reactor). Chemical species released from the damaged core material may lead to further explosive reactions (steam or hydrogen) which may further damage the reactor.
(References have been removed from the quoted texts.)
In short, the reactors will shut down, but if there's no-one left to keep the cooling going properly for long enough, then it's quite likely there will be some serious accidents.
So there's a very good chance that large-scale power generation will be shut down rather quickly, within a few days or weeks. Not necessarily broken and unrepairable, though.