The rear derailleur wakes up first as it is the “brain”. AFAIK the shift buttons wake up once you press them. If you have a power meter, it could wake up, too.
The LBS told me when they handed me my new road bike with Force eTap AXS that the battery latch is prone to breaking and apparently cannot be replaced separately. That’s why I should put in the red battery cover on the rear derailleur when I take out the battery.