US BWR worker/nuclear engineer here. Our tornado/high winds offnormal procedure does NOT require us to shut the facility down. I do know for plants that are on costal regions, sustained hurricane force winds require the plant to shut down. My plant is not anywhere near an ocean though. When we do have tornado winds, or if there are warnings/watches, we protect critical equipment in the plant and halt/back out of maintenance which could affect safety systems required to ensure safe shutdown.
Browns ferry in the last year or two was on diesels for a few days when they lost like 9 offsite power connections. It was considered impossible for them to lose all offsite power, so it was kind of crazy.
By shut down, I mean a manual scram (or insertion of all control rods).
To the best of my knowledge, there is no requirement that a plant shutdown for tornados, but if extensive damage is expected then a shutdown would be prudent (and would occur regardless due to a load reject, loss of auxiliary/standby power, or failure of circulating water pumps).
During a large tornado like event, depending on what/where the tornado hits, there may be reduced electrical demand on the grid, this could cause the plant to have to reduce output and potentially take the turbine/generator set offline. During Hurricane Sandy, I know that several plants had reduced power, some very deeply, due to Volts/hertz or reactive generator voltage limits (there wasn't enough demand on the grid to satisfy all the power going out), and I know one plant who was very close to de-syncing their main generator as they were close to the high voltage grid/generator trip setpoint.
As astronuc stated, if your fuel is preconditioned, you can go down/up rather quickly. One thing to remember is that fuel preconditioning starts to wear off after 4 hours, so the timeline for a rapid load reduction and restoration is pretty small, and in most cases would still likely require some preconditioning or a separate rod sequence (I'm a BWR nuke) to return to full power operation in a timely fashion.
There really aren't benefits from a risk or operating perspective to shutting down for a tornado ahead of time. Typically tornados are events that occur quickly, so you wouldn't be able to anticipate a tornado heading towards the plant, shut down, and cool down a reactor from hot standby to cold shutdown prior to the tornado hitting(which is a much 'safer' condition to be in). So that means regardless of whether you scram early or not, you have a hot core with a large decay heat load.
If you scram early, that tornado may never even come close to the plant, and now you just put the plant into a transient (and any transient, even a reactor scram, has risk associated with it), and also put thermal cycles on all your equipment, you also now have a reportable event you didnt need, it counts towards your INPO rating, and you challenge your plant systems needlessly. If I scram early, the only benefit is that I can start my cooldown on non-safety systems (condenser with steam bypass). But it wouldn't matter, because if/when the tornado causes a loss of offsite power I still would need my diesel generators to start and I would have to use SRVs, RCIC/HPCI/IC for pressure control (or atmospheric dumps for PWRs).
The main risk with tornados is station blackout (loss of offsite power and a failure of all diesel generators responsible for decay heat removal), which the US has 4 hour coping requirements and blackstart plans/generators to mitigate (as well as the new FLEX/fukushima requirements and the old b.5.b requirements). The next large risk is damage to plant auxiliary systems, usually emergency service water pumps or other things which are more 'external' to the plant, but it could also be station vents, external water tanks (like those used for RCIC/HPCI), main transformers.
Anyways that's my blurb.