As it turns out, the reason there's no extra loading screen is because all three stages actually exist within the same map space, just on radically different levels. Thus, all three rounds are actually pre-loaded during the main bulk loading phase.
The method was discovered thanks to some fancy footwork between hulkman503, a friend and a whole lot of Mercy shenanigans.
The video below explains how they achieved this:
It's by no means an easy trick to pull off, not least because during the travelling phase you need to avoid touching any of the invisible patches that trigger death.
Given the difficulty of pulling this maneuvre off, you shouldn't expect to see too many of your game's ruined by teammates going off on an adventure. We do, however, expect Blizzard to fix this glitch in pretty short order.