im pretty sure it takes 10 minutes before you're even frozen
Quite possibly. I'd be surprised if you can get out of the vault within 20 minutes.
What with the time it takes to design a character (should you do so), having to interact with things around the house, talking to the vault tec guy, running to the vault, getting into the cryo chamber, and waiting for the story during, then actually escaping.
It's probably quicker than Fallout 3's start however.