Download the upscaled texture pack and the FMV pack (for videos) from moddb

Main mod:
https://www.moddb.com/mods/the-longest-journey-hd/downloads/tljhd

FMV pack for 1080p:
https://www.moddb.com/mods/the-longest-journey-hd/downloads/the-longest-journey-hd-fmv-pack

Install The Longest Journey.

Extract the contents of 'TLJHD.zip' and 'TLJHD_FMV.zip' to the game installation directory. If asked, overwrite all existing files. Make sure that the path to the games looks like this:
/.local/share/Steam/steamapps/common/The Longest Journey

Install ScummVM from https://snapcraft.io/scummvm

(I used the latest stable version)

Click Add game... and choose the game (not the mod!) directory. It should be /.local/share/Steam/steamapps/common/The Longest Journey


The graphics tab should pop up automatically, if it doesn't, click on Options, go to the Graphics tab, select Override global graphic settings and Fullscreen mode.

Enable anti-aliasing for the 3D models and select the level of Anti-aliasing in the drop-down list. I use 8x Anti-aliasing.

After all of this, Click OK

You can now launch the game via the ResidualVM GUI. Should be in your launchpad under Games.