Electric toilets??
the trigger mechanism for the flush, is my guess. you can go, but you can't flush.
the water pumps are electric
Could be several problems. One without power water pressure may be low or non-existant as the pumps are not working.
Second, have you not ever seen them fancy self flushing toilets? You stand up and the electric eye senses you are not sitting on the toilet and flushes it?
Electric only terlets were a problem in the last NYC power failure, too.
I've seen newer ones with a redundant manual flush button, and ones with a water flow powered generator to avoid this problem.