Nuclear engineer here
The plant actually runs on generator power under normal conditions.
Nuclear plants have 4 AC power sources. The normal source is taking generator power BEFORE it goes out to the power grid in through the auxiliary transformers and then using internally for 4160 and 6900V power. Because this power hasn't gone to the grid yet, we don't "pay" for it. Additionally, when we are shut down, we can disconnect the generator and backfeed power in through the aux. transformers for power. This is typically an emergency/contingency action or an outage action to allow us to work on the reseve power system.
The standby source comes in from a different grid (or a different part of the same grid), and comes in from the reserve auxiliary transformers (sometimes called startup transformers). Because this is bringing power in from the grid, we "pay" for it (we get billed by the grid).
The emergency reserve transformer (sometimes called backup transformers) comes from a completey different grid than everything else. They power ONLY safety systems. Normal systems cannot use it.
The diesel generators are safety seismic and environmentally designed backup power systems. There is 1 DG for each primary safety division which has a decay heat removal function, and an additional DG for coolant injection. Most plants also have a fourth or fifth DG for DC power chargers only. There is enough fuel on site for a minimum of 1 week for all generators running 2% above maximum theoretical load of all equipment under worst case design conditions. The reality is you can probably get another 2-3 days past that since it assumes that like, air coolers and air heater are both on at the same time in the same area, and once you've stabilized an accident or emergency condition you can put most of the redundant safety systems into standby to conserve fuel.