[jaak]

The Tesla App seems to be divided into convenience features (which use an Internet connection) and a key (which uses Bluetooth).

The key part does not rely on Tesla's servers to operate (except probably for initial setup?). I had no problem unlocking or starting my Tesla during the outage.

Sadly, the App (convenience part) isn't 100% reliable. It occasionally fails to connect to the car and the only way to get it to work is to force close it. However, I have never had the key part fail to work.