Suppose a fault in the reactor occurred. The reactor would still need to be cooled but could not be run to generate power to cool it, so an EXTREMELY BAD SITUATION would develop. Therefore you have to have a backup system for power, and although it may appear paradoxical, offsite power for control (or backup onsite) is the safest way to do it.
The reactors will authomatically SCRAM, which stops fission, but the decay of the radioactive elements from the fission still produces considerable heat, although much less than during fission, and the heat drops off quite rapidly over a week.
There are also multiple cooling circuits so that if a fault in one occurs the reactor cooling will continue.
Some types of reactors have a system where leftover steam can be recycled to a system that circulates water too, but obviously that's temporary.
The power usually goes into batteries which directly feed power to the reactor's control systems. The batteries have a few hours of time left and the onsite power swings in to replace power flow to the batteries.
Specific info on Surrey here at a blog I found while googling:
http://atomicpowerreview.blogspot.com/2011/04/surry-power-station-tornado-touchdown.htmlWaterford, which is I think about 25 miles from New Orleans, was without power for over a week from Katrina. And they were hit by the hurricane. They shut down fission proactively and ran fine on the onsite generators.
http://www.google.com/url?sa=t&source=web&cd=3&sqi=2&ved=0CCUQFjAC&url=http%3A%2F%2Fhome.engineering.iastate.edu%2F~jdm%2Fkatrina%2FData%2FElectricity%2FElectricity%2520Generation%2FNRC%2520reports%2520on%2520Waterford%2520nuclear%2520power%2520plant.doc&rct=j&q=Waterford%20Nuclear%20power%20plant%20Katrina&ei=jsmsTf6mIIfMgQfpsfmSDA&usg=AFQjCNEzV745gMbKnZI9pGP8Dxvang-bbQ&cad=rja