The collapse would be pretty dramatic and cause it’s own issues just from a construction/accident response. From a system response point of view, it would all depend on the system design. Usually cooling towers are included as part of the secondary cooling loop, removing excess heat via steam.
Most nuclear sites will have redundant & backup systems that will take over if the normal system fails to work. It will be a priority to get the cooling system working properly to prevent the nuclear fuel from overheating and potentially start to melt.
The current nuclear fleet in the UK doesn’t use cooling towers, they pull water in from the sea and use that for cooling before discharging it back into the sea. But cooling towers still exist for some coal or biomass power stations (most I think are scheduled for demolition) and for different industries like in manufacturing or chemical factory’s.
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