When you're connected to City water, then your lines pressurize and water flows normally out of your faucets. That would suggest that all of your plumbing downstream of the pump is in good shape.
When you are not connected to City water, then you pump needs to turn on to pull water from your tank. I assume you know where the pump is. Can you tell if it is working? When you apply power, you should hear the pump running. Frequently there is a valve right after the pump used to drain your system. Some call it a low point valve also. If you open that valve when the pump is running, does the water come out with gusto? If it does, then you know your pump is primed and pumping. The problem is further downstream.
I assume you filled your water heater when connected to City water. Otherwise your pump could be spending time pushing water from your tank into your empty water heater.
Some pumps have a valve that direct the output of the pump to the downstream plumbing or back into the freshwater tank. I've never understood why they have it, but if you do, make certain your water is flowing downstream to faucets and not back to the tank.
Hope some of this helps or gives you clues... I'm curious what you find to be the answer!
