Microsoft's Larry Hryb has posted a short clip showcasing how long it takes an Xbox One to boot up from a Kinect voice command.
I timed it at 13.3 seconds. That's going from the moment he stops talking to the point where the dashboard pops up ready for user input, which seems to me the most practical passage of time. (Feel free to chime in with your own timings if my trigger finger is sloppy!)
Hryb says it "happens faster than it takes to find your controller". I don't know, I timed that too; it was around four seconds.
On the bright side, it then took my 2007 Elite 360 around 117 minutes to boot up once I'd hit the button, so this seems like progress from where I'm sitting.
Note: this is one console, using Kinect as a boot-up. Don't go taking this to the bank that all consoles will boot in this time under all circumstances.
You can watch the whole sequence below. Power up (and sign in) to #XboxOne with Kinect. [Xbox]