What happens to the flow rate of water through a hydro electric generator
Since the grid is electrically much "larger" than any individual generator, the generator will continue to run at the same speed.  What changes is the phase angle between voltage and current, and the magnitude of the current.  There can never be more real power out of the generator than what is coming in via the water flow.  In the extreme (wouldn't happen in any reasonable normal operation), the grid drives the generator so that the turbine starts pumping the water instead of the water moving the turbine.

There are also control systems around hydro generators at several levels.  The fastest response is by controlling the field current.  Then there is a slower control system that adjusts valves for the optimal water flow.  These systems largely operate autonomously already (once enabled and configured by the humans), so until something drastic happened, a hydro plant would continue working properly as load changed without humans tweaking knobs.