You might not realize it but a reactor plant is not something that you can just turn on and off like a diesel. We would spend about 8 -12 hours MINIMUM doing prestartup checklists and a few hours to actually take the reactor critical and bring systems online.
The Americans did try something like this with the USS Tullibee. It was turbo electric drive with a big honkin electric motor driving the shaft directly. The power requirements are a LOT different when you are just driving turbine generators and not the engines. It supposedly cut the noise level down propulsion plant noise significantly but due to the way it was designed it was not very fast.
Oh, and american subs DO have batteries. BIG ones, but they are a backup for a backup of the reactor plant. (yes, I said backup twice for a reason)
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