People have been talking about using pumps and computers to ensure that hot water comes out the tap at the moment's notice. (for example, hot water home run return mixing)
You don't need a pump, merely a pressure differential. You could do this using a thermostatic valve, a pressure reducing valve, two check valves and a pressure vessel. (A clever person could make all of these in a single HDPE unit)
Hot tap turns on, water flows from hot tank to cold pressure vessel, water warms up, thermostatic valve opens and out comes water.
Cold tap turns on, water flows out from pressure vessel until below pressure reducing valve pressure, flows from mains. The cold water could be dumped into a non picky load such as toilet cistern.
(personally I'm a little dubious of the merit of the problem)