It seems that this achievement has been broken for some time on the current (oct. 2023) version of the game. If you still wanna go for it, you need to downgrade your game version to an older, working one.
Some guides and information are available in the comments below. Thanks to Brandonius for sharing the tips.

First, you need to finish the game. After that, when you restart the game, you will spawn in the room between the second water hub and the space levels. This is where we start.

After you have finished and restarted the game, take the big door at the top of the room that leads to the second water hub. You don't need to do the Space part again.

Play the three levels in this part: Swamp #2, Wetlands, Winter. The order doesn't matter. When you enter them, the elevator will take you at the end of the level, you just need to get back to the other elevator.

Then, go to the first water hub, and play Midnight, Morning and Swamp #1 as well (no matter the order).

Finally, go back to the intro level, and play it to the first screen.

You did it! Congrats!

• Swamp #2 and Midnight are really easy to do backwards because all the wormkeys have been collected so you just need to take the portal to get to the first elevator.
  
• Wetlands and Swamp #1 are quite easy too, you don't need to solve the puzzle on Wetlands again, and there is no specific challenge in playing Swamp #1 backwards.
  
• Winter and Morning can be tricky sometimes, but everything has been well thought, and you don't need amazing new techniques to go through them.
  
• You don't really need to do first the three levels of the second hub, then the three levels of the first hub. It's just the obvious way to do it :) The intro level must be done last, though.

Now go work on your Minimalist and Speedrun achievements!