Chances are nobody knew the answer , best bet would be to contact nest support directly at this link, as we are only a community forum. From your original post I see the pilot light goes out with the nest thermostat, my suggestion would be out the old thermostat back in until you can spend the time with the support team as they may have more information for you. A C wire is a definite requirement for flawless use in most cases for sure.
https://support.google.com/googlenest/gethelp?hl=en
