AC won't turn on after replacing thermostat batteries
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.