Yes. I just received an email from Tesla Customer Support via the online support request. They confirmed that both HW2.0 and HW2.5 cars will get the new APE3.0 autopilot computer. They specifically stated that no sensors, radar, cameras, or other equipment will be upgraded.
Now that I have in writing that my 2.0 car will be retrofitted (at some future, distant point I'm sure), I'm going to plop down the $2k knowing prices are going back up Monday.
the pop up is caused by an automated job. There's nothing manual here. A bot logs into the car and issues a gateway configuration change and then the MCU displays that FSD has been unlocked (though no other user facing FSD features or changes actually exist beyond the pop up). If you never got the pop up, I'd call support and verify the change went to the right car.