Normally, when playing the game and loading the next level after getting on the bus, you will see a black screen, then a white, pixellated message saying "World <level number> <level name> Start!". You can mash buttons rapidly to skip this message, however this often results in a strange occurrence in which the game loads the next level, but does not allow you to progress, before the screen reloads and you can go. This is still a time save compared to the normal loading, however a much more efficient process is easily available:

After completing a level, and as the bus takes Bernie off screen, hold your right analog stick in any direction (I believe this only works on a gamepad). If you get the trick correctly, you'll not only completely skip the level text, but you'll also cleanly fade into the next level, saving the maximum amount of time possible.