Looking at your screenshot, put the HUD mod inside that skins folder as it is alongside gzdoom.exe. Then launch the executable with your shortcut.

Can you post the contents of the shortcut? So that we can see what parameters you are using to load it up with Brutal Doom. I mentioned that I use a batch file, so for me the command I use is:

START gzdoom.exe -iwad %iwad% -file %addons% -config %config% -savedir %savedir%

Here I use a number of variables that are set depending on the options that I select in my batch script user prompts.

%iwad% is the Doom WAD to use (e.g. DOOM.WAD)
%addons% is my list of addons to use, e.g. Brutalv20b.pk3
%config% is the path to the config file to use (as I use a separate ini file for each mod/TC)
%savedir% is the path to the save folder to use (as again I use a separate save folder for each mod/TC)

If you are using a standard config and save folder you can ignore these parameters, so launching Brutal Doom would just be:

START gzdoom.exe -iwad doom.wad -file brutalv20b.pk3

The HUD mod being in the Skins folder will then launch automatically.

Ah, I see, so you have a separate folder for each of the Doom games from the GOG installations and you've put a separate instance of GZDoom in each one. As such, you need the HUD mod in every skins folder, as each GZDoom is a separate instance.

By associating GZDoom to the pk3 file type, I'm wondering whether it is calling the wrong GZDoom installation, as you have multiple GZDoom installations, one in each Doom game folder. So it might be looking for the HUD mod in the wrong place. I wouldn't do it this way. If you really wanted to have an easy way to load the Brutal Doom pk3, you can just drag it onto the gzdoom.exe in the folder and it will then start up, but may prompt you for the IWAD if it detects all the different Doom installations.

As for No Rest for the Living. I would still use the non-BFG Doom games (as they don't have the unnecessary changes made to them and I've read about people having issues with them, the non-BFG versions are preferred). There are plenty of addons available that have No Rest for the Living. NRFTL+ is one that I've used that has map fixes to fix bugs in the original release, and it shows up as a separate campaign for Doom 2 if you load it up with the DOOM2.WAD file.

It must have been loading the original BFG GZ instance: I added a new skins folder to the BFG WADS one and it now works!

I would say that by adding a file type association for PK3 files that your Windows always refers to a specific GZDoom installation, then, as I suspected, which isn't ideal. Personally, I'd break the association and use a shortcut to specify the GZDoom executable to use with the Brutal Doom PK3 given as an argument for the -file parameter. And if you want to run multiple mods at the same time (such as Brutal Doom with a custom map pack) then double-clicking the PK3 won't be of any use as that would only load the double-clicked PK3, not all of the ones wanted.

Protip for later: No Rest For the Living is not Doom.wad. It is Nerve.wad: https://doomwiki.org/wiki/NERVE.WAD

It sounds picky, but this is important when asking for this sort of thing. It's definitely BD compatible though.