You can't unless you apply steam patched SeriousSam.exe to gog version that you can find on community websites.

Prease read previous posts.

Yup. When Croteam made an update with fixed widescreen they only released it on Steam. GOG still has the original version of the game. No clue why they didn't update it on GOG.

Well, then I guess I will try to contact them and ask them about the missing upate on GOG. Maybe we'll get lucky.

As promised I asked this question on their discord and I have been told that the classic release never had official HD support. The version which supports HD is calles Serious Sam HD and it is a different game. If this is the case we aren't talking about a missing update at all.

They probably misunderstood you.
In 2015 our russian mate Slawww (he is in contact with Croteam) made an update for Steam version to restore multiplayer (changed masterserver from GameSpy to Amsterdam42), add support for 16:9 and 16:10 resolution with correct weapon FoV and fix Serious Editor compatibility with 64-bit systems.

So the classic version on Steam also is HD?
Apparently it is - the people on discord just confirmed it. However so far I did not speak to the devs. If theyx don't reply I will try different contact possibilities as well.

Alright, so Serious Sam HD TFE/TSE are remakes of the original games. They have different textures, different models, different engine, even different level design in some places. You could consider them separate games in the series. Steam has an update to Serious Sam "Classic" (the original games), which fixes widescreen resolutions.

The original game only allows you to select 4:3 (and 5:4, I think) resolutions in the settings. The Steam version fixes this by adding an aspect ratio selector: you can select 16:9, and the widescreen resolutions become available to you. You can set the resolution manually in either version by editing PersistentSymbols.ini, but this creates problems with FOV.

The GOG/Retail version keeps horizontal FOV fixed, while adjusting the vertical FOV to match the aspect ratio. This means that as your aspect ratio increases (becomes more wide), the vertical FOV gets narrower. This technique is called Vert-, and it was used in many games from that time period. This makes the game feel "zoomed-in" in 16:9. The Steam version fixes this by changing the FOV scaling to Hor+, which is the exact opposite (vertical FOV is fixed, horizontal FOV is adjustable). You could also "fix" this in the old version by adjusting the FOV with plr_fFOV command, but there's a caveat: when playing scorematch/fragmatch multiplayer, the plr_fFOV command is ignored, and the game acts as if it's set to 90. I think this was done to prevent people from having competitive advantage by playing with plr_fFOV=160. Now you could get that advantage by playing the Steam version in widescreen (which, I guess is a downside of this fix :-)

Additionally, the old version relied on GameSpy for the multiplayer server list, which died in 2014. Steam replaces that with 333networks, which is a replacement server that can be used with most GameSpy games. Another thing that Steam fixes is the editors, which used to simply crash on 64-bit systems.

I've attached a couple of comparison screenshots with GOG on the top/left, and Steam on the bottom/right.

Regarding the missing update for Serious Sam: The First Encounter here on GOG.
After asking two times on discord and getting my mail back from the website because it could not be delivered I finally posted a DM to the mod on discord. The mod did not react but he did not have any problems posting more announcements for the new Serious Sam game so I complained and that's when I got the following reply:

"I'm sorry, I don't know you. I'm not obligated to reply to you privately. I am also allowed to miss your messages when they come in because I've got work to do. Maybe a gentle ping would do you better than straight up rude comments. There are currently no plans on updating SS TFE on GOG."

Croteam cannot expect that I will ever buy a single game from them in the future with this attitude.

I wouldn't expect a discord mod to be a Croteam employee.

This topic is about the in-game widescreen fix, which works if done correctly, has the true widescreen resolution, not the distorted stretched out picture. This has nothing to do with the third party programs.

More info to go with this is "...\Scripts\Game_startup.ini" is like the old "autoexec.cfg" from the old Quake engines and any commands in it are executed at every game launch. Here's my list of most important settings/fixes:

persistent extern INDEX sam_iScreenSizeI=(INDEX)1920;
persistent extern INDEX sam_iScreenSizeJ=(INDEX)1080;
persistent extern user FLOAT plr_fFOV=(FLOAT)104;
persistent extern user INDEX hud_bCrosshairFixed=(INDEX)1;
persistent extern user INDEX hud_bCrosshairColoring=(INDEX)1;
persistent extern user FLOAT hud_fCrosshairScale=(FLOAT)0.5;
persistent extern user FLOAT hud_fCrosshairRatio=(FLOAT)0.7;
persistent extern user FLOAT hud_fScaling=(FLOAT)0.5;
persistent extern user FLOAT gfx_fEnvParticlesDensity=(FLOAT)0.9;
persistent extern user FLOAT gfx_fEnvParticlesRange=(FLOAT)1.1;
persistent extern user INDEX gap_iRefreshRate=(INDEX)120;

Also, "Sharp Turning" = On, in the options fixes horrible mouse lag. I think the mathematically "correct" 16:9 plr_fFOV is 106.26, but that seems just a bit fish-eyed to me, so tweak to your liking...probably anywhere from 102 - 108.

Regarding all this, be it known that there are source ports for TFE and TSE, onto modern graphical APIs, no less.

That all looks very nice, but No Windows version? Seems to be Linux, MACOS via Steam only? Very confusing.

Edit: There is Windows versions, just had to look closer at the file list. Works really nicely on Linux though! Thanks. Just a matter of copying over all the assets into the pre-compiled engine.

On Windows, it runs very "choppy" in some areas, even though maintaining a capped 120 FPS (frame skipping?), both the Directx (with bin-tfe, bin-tse files), and the Vulkan port, but on Linux the Vulkan port is smooove!