Think about something said earlier about the clock starting/not starting depending on whether the Team A player was upright v. on the ground on the on-side kick at the end of the game...
If the Team A player cleanly possesses the kick, the clock does not start PERIOD. Team A can't advance a recovered kick, only possess it, so whether the player is upright or down shouldn't matter.
If the kick went 10 yards and the Team A player muffed the kick, the clock would start and in this case expire with only 1 second remaining when the play started.
The interesting "what if" scenario would be an illegal touch before 10 yards, possession by Team B and B is down at the 10 yard line... Game over, as an illegal touch is a violation and does not give Team B an untimed down.