The good news is that this can be done on I'm Too Young To Die difficulty and only counts the time from the first 3 episodes.
Here are some speedrun guides. These were not made by me. The first is Episode 1 done in 8 minutes by a YouTube account called Dinglemaniac:
Episode 2 by Lammah on YouTube. This guy does it on Nightmare in under 6 minutes! So you will find it even easier:
Episode 3 by Zero Master in under 4 minutes:
Needless to say you have a lot of room for error. The best advice is to have one save right at the start of a level and another for during the level when you think you're doing well. I did every level under par time and got the achievement so if you're on the same track you'll be fine.