Recently we had Rinnai heating water system installed in the house and now we are experiencing fluctuating water pressure on the cold as well as hot side. We are on a private well .

Check your pressure gauge on the well as you are using water. It should SLOWLY go down until the pump kicks in, and then, in all likelihood, go up even more slowly until the pump cuts off. If the cycle is faster, in a few seconds each way, then your tank is probably the culprit.