The reactor in a nuclear submarine does not need air to operate. It produces heat to make steam like a large kettle and this drives turbines, which turn the propeller shaft. The steam is also used to operate turbine generators that produce electrical power for use on the boat.
Modern submarines have a nuclear reactor as the main source of propulsion and a diesel generator for secondary propulsion.
Nuclear submarines which can remain submerged for as long as necessary without surfacing have special air purification systems installed. These consist of Carbon Dioxide Scrubbers to remove the dangerous CO2 gas from the air and pump it overboard and machines called Electrolysers that produce oxygen with hydrogen from seawater. The oxygen is kept and the unwanted hydrogen is pumped overboard.
Surprisingly, when submariners emerge from a patrol, the 'fresh' air we breathe smells quite foul to them for a time compared with the purified air they have been breathing onboard!
