I’ll have to give you a theoretical answer because the realities of IFR clearances and ground traffic get in the way of the “airborne” limitation in your question.
But if you had one parked at the end of a runway and had the enroute clearance so you wouldn’t have to sit on the ground with an ATC hold, I could have gotten airborne in five minutes from dead cold. I’ve come close to that even with the realities of other commercial traffic.
You’ll note that one answer below says that it would take 20 to 45 minutes to get a Boeing started up and airborne. I’m going to guess that that writer has never fired up a cold Boeing because if he took that long, he wouldn’t last long in commercial airline service.
THE EASTERN SHUTTLE STANDBY JET
Back in the days of the Eastern Shuttle service between La Guardia and Boston Logan and Washington National, we offered service every hour-on-the-hour with no reservations and a guaranteed seat for everyone who walked into the gate by departure time.
To do that, we had to have a backup jet standing by. I’m a lazy ass when I can get away with it so I bid the shuttle standby because I only had to fly once or twice a week instead of making two round trips per day.
The standby was supposed to leave 10 minutes after the hour if the prime on-the-hour flight filled up. The reality is that most times we took longer than 10 minutes to get going because of the time it took to board passengers—and since they were streaming in, we’d let them keep coming.
But quite often we went empty. That happened when the New York prime had filled up and they used the standby jet, which meant they wouldn’t have a standby jet the next hour unless we went up there. It was called a crossover in which a Washington crew flew up empty to take over New York standby duty while the New York crew took our place.
In those cases, we could easily be fired up and taxiing in five to 10 minutes. Again, the “airborne” part of your question obscures the truth because we could easily sit for 10 minutes at the end of the runway waiting our turn to take off.
The bottom line is that one can be fired up cold and take off in five minutes—and 10 minutes would be really easy.