Either your water heater input valve or output valve is not open, or something is blocking the flow of water from the water heater--sometimes there's a check valve built into the water heater, and that could be failing to let water through if it were stuck shut.
Close the bypass valve (so that the water from the faucet stops). Turn the other valves to their opposite position, one at a time, and see if the water flows. If not, I'd detach the connections to the water heater and see what's going on there. (Obviously, in this last case, the water system and water heater should be drained unless you fancy a flood.)