I normally power off via an on/off switch when I leave the car. What I did was power on, leave it on and left the car. I gave it enough time where I no longer saw the T2C connected to my phone. When I jumped back in the car I was connected to my home Wi-Fi and the Tesla would not automatically switch to the T2C even though I could see the Wi-Fi connection. I blocked my Tesla from connecting to my home network and repeated the same steps. When I got to the car it was connected to LTE instead of my home Wi-Fi and it automatically switched to the T2C and the CarPlay screen appeared in 49-52 secs.
Thank you for checking. My Tesla Android is connected to my Tesla in a way that the power is shut off every time I close the door and leave the car. I prefer this way so there is one less item draining the power from the Tesla. So every time I enter the car the Tesla Android has to start a power up cycle. I would install any device including this T2C device the same way. In this situation, in your case you can shut it off and turn it on again with your on/off switch - what is the difference between the T2C and the Tesla Android for the timing to fully power up and have the web page available? Thank you again for checking.