/!\ Warning /!\
The following has been tested on the steam version of Civilization III Complete, not on the GOG version, but I suspect that it should also work. Tell me if it doesn't.
Also, I tested this on a Windows 7 64-bits, so I don't guarantee that it will work on a Windows 10 or 11.
/!\ End of warning /!\

I own the steam version of Civilization III Complete and it's supposed to include the base game and the Play the World expansion, but I noticed that you couldn't launch them, only the Conquests executable is present, so you can't play the game as it initially was, without the slight gameplay changes introduced with Conquests. And from what I read, the GOG version also has the same flaw.

So I did some research and I found out how to make the vanilla & the Play the World extension working. Here is how to do it.

Both former versions are actually present (all the datas / assets are there), only the game's executables are missing. So the first thing to do is to find a NoCD exe for each and add them to the game's directories. Sorry, I can't help you to find them and put any link in here, but there are places where to find them on the net. It's important to get the good version of the NoCD though :
Vanilla Civ III's latest version is 1.29f
Play the World's latest version is 1.27f

Put the vanilla's exe straight in the game's directory (for example something like "D:\games\Sid Meier's Civilization III Complete")
Put the Play the World's exe in the civ3PTW subdirectory (ex : "D:\games\Sid Meier's Civilization III Complete\civ3PTW")

Then, you may try to launch either one, but it is likely to fail because there will probably be several missing registry keys. There they are :

Open regedit (start, run, regedit) and go to "HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Infogrames Interactive\Civilization III"
There, you should have 2 string values : one called "CD_Path" and the other called "install_path", both pointing to your Civ III folder (ex : "D:\games\Sid Meier's Civilization III Complete\" or whatever is your folder), so create any one who is missing, especially the "CD_path" one.

Then go to "HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Infogrames\Civ3PTW"
There, you should create another string value called "install_path" and set it to your PTW folder (ex : "D:\games\Sid Meier's Civilization III Complete\civ3PTW")

Quit regedit, and now it should be working for the 3 versions of the game.

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I don't know about the GOG version, but on the steam version, there is one last issue to solve : the menus in Play the World are completely messed up. This is because of a modified labels.txt in the civ3PTW\Text subdirectory. You have to open the labels.txt file in notepad, search for "PING", and this is what you'll find :

PING
Unknown
Host

Delete the "Unknown" line so that there is just :

PING
Host

Save the file, and now the menus will be ok.

But since this seems to be a steam-specific modified file, this probably won't be needed to do on the GOG version.

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After that, all 3 of the games should be working fine. Only the intro videos of the vanilla & Play the World are not the correct ones in steam, they still are the Civ III Complete one, I don't know if it is the same on the GOG version, but that's not a big deal anyway.

Enjoy.