we replaced the batteries on our thermostat after they began flashing for a few minutes. We'd been using the heater up to that point. After replacing the batteries, the heater does not turn on anymore. We can hear the click indicating it's trying but nothing is happening.
If your thermostat requires you to remove it from the wall to replace the batteries, with a wall plate left affixed to the wall, then it sounds like you didn't properly reattach it to the wall plate. There will be electrical contacts between the two halves, often long pins, that need to properly align. A bent pin, incomplete attachment, dirt on the contacts, etc, can all result in the symptoms you describe.
The clicking is the batteries turning the Stat on or off and/or relative Stat functions. If you run a common wire to the Stat you can do away with the batteries. Remove Stat and check for a voltage reading between R and any of the other wires, you should read 24 to 28 volts. If not then that's your problem, no control volts!