Best I can do for you since I don't have a Steam or GOG version of Fallout 3 (I don't buy games I already purchased). Bethesda really should have released a patch for the retail ver to remove GFWL.
For Fallout 3 that came on DVD
1. Install from media (You can do this on XP or on a newer version of Windows)
2. Patch to v1.7 (You need to do this since it removes copy protection from the launcher otherwise you'll have to hunt down a crack )
3. Try the launcher, If the launcher doesn't recognize that your game is installed you'll need to make sure the registry is correct.
a. 32bit windows Copy the following into a .reg file import with regedit /s. Edit with your installed location.
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\Software\Bethesda Softworks\Fallout3]
"Installed Path"="C:\\Program Files (x86)\\Bethesda Softworks\\Fallout 3\\"
b. 64bit windows Copy the following into a .reg file import with regedit /s. Edit with your installed location.
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\Software\Wow6432Node\Bethesda Softworks\Fallout3]
"Installed Path"="C:\\Program Files (x86)\\Bethesda Softworks\\Fallout 3\\"
4. Download xliveless and place the config file and .dll in the game directory so you don't have to install GFWL
5. In the "Fallout_default file in the game directory make the following changes:
bUseThreadedAI=1
iNumHWThreads=2
6. The above was tested on Windows XP.
7. If you want newer Fallout 3 executables (they remove the GFWL requirement) then they may not work on Windows XP.
For instance the the idiot that compiled the Epic version of Fallout 3 compiled the Fallout executables with a newer version of Visual Studio which does not work on Windows XP and requires additions VS .dlls.
It's likely that the GOG and Steam executables work fine as long as steamless is used to remove the Steam requirement from the Steam version. For instance the Steam version of Fallout New Vegas works fine on XP.
You can also try this but I've never tested it (Unknown if it removes GFWL if you are already at 1.7.03):
https://www.nexusmods.com/fallout3/mods/24913 /EDIT v1.7.0.4 Steam ver requires Vista or above. The v1.7.0.3 Steam ver should be fine.
Assuming GOG version is 1.7.0.3 then should be XP compatible
Note that any ver that has v1.7.0.4 can just swap out the executables with the v1.7.0.3 ones to be compatible.
// The Anniversary patcher just uses xdelta to revert to the previous executable version as well as the additional tweaks so nothing special but it will give you a v1.7.0.3 executable that doesn't require xliveless to disable GFWL
///So assuming GOG is at v1.7.0.3, if they do "upgrade" to v1.7.0.4 then they need to make sure that the executables are compiled with at the most VS2017, if they don't care then the bare minimum they should do is have the v1.7.0.3 executables in addition to the v1.7.0.4 executables especially for mods.
////Newer versions TTW are supposedly incompatible with XP due to 3rd party addons being incompatibles so for those wanting to run it would either have to use an old ver of TTW or seing if those addons can be earlier versions or swapped out.
/////If the GOG version requires GOG galaxy then the GOG galaxy .dll requires Vista so you'll still need to swap out executables with or use the patcher.