Well, we don't know exactly why the reactor was manually shut down. Did the person actually know that the control rods had dropped? Or was it reported by sensors? I suspect the procedure us to initiate shutdown in response to any reports of uncommanded rod behavior. I doubt they would wait around to see if this was a sensor or reporting system error.
The dropped rods already shut down the reactor in the sense that it is sub critical. It still has enough leftover heat that it will continue to produce steam for a while, but it would have been almost instantly clear to the operators what had happened.
[Correction] as noted below, this type of reactor -- unlike US Navy ones -- can remain critical even with dropped rods. So a better statement here would be that the operators would immediately notice a reduction in criticality. thus power and heat. Then it's a process/policy question about shutdown timing.
Navy reactors are physically tiny. A rod in a reactor the size of a garbage can means much more than a rod in a reactor the size of a studio apartment.