Nest thermostats can function without Internet. You can walk up to it and adjust the temperature just like any old thermostat.

I live in a house with a Nest thermostat and Nest Secure and Nest x Yale door locks. The AC is on just fine, and the door unlocked just fine. (The door lock doesn't require an internet connection, unless you've enabled privacy mode for some reason.)

I thought there were versions that don't work offline; my bad if this is not the case.

Internet isn't the issue, it's the endpoint (Google)

It's reasonably easy to run a router/gateway that has a 4G backup to get the ping out. Whether the ping works...