Binweasel: I think I found it! Try C:\Program Files (x86)\GOG.com\Fallout\DATA\SAVEGAME\
Then each save is its own folder, SLOT01, SLOT02...
blackspy025: It did bring me to an existing file, but it's empty. In my game, it does have all the save slots ready and working though, so I don't know what could be happening. The only files in Fallout>data that isn't empty is Scripts, sound and text. The maps and savegame files are empty. :/
If the save folders at C:\Program Files (x86)\GOG.com\Fallout\DATA\SAVEGAME\ are empty, it means you probably haven't been running the game as administrator.
Without admin permissions, the game can't write to its own directory, so it uses some kind of jury-rigged method of saving without creating new files (this is true of many GOG games).
I don't know how it does this exactly, but I do know that it makes your saves _highly_ prone to corruption. It also means there are probably no save files as such to be found. If I had to guess, I'd say the game probably writes save data into existing game files.
Your best bet might be to copy the entire installation of Fallout to your laptop. To do this, I would install Fallout on the laptop first (install into the default path), then replace as many game files as possible with those from your PC installation. If you're lucky, you will have imported your save data. Run the game (in this case, NOT as admin) and hopefully your saves will be there.
I don't know if you're going to find your saves or not (I hope you do!), but there is a lesson to be learned: always, always, ALWAYS run GOG games as admin, either by right-clicking the game's shortcut and choosing 1. "Run as Administrator", or 2. "Properties", going to the "Compatibility" tab, and checking the "Run as Administrator" box. Alternatively, I think you can install outside of the program files (x86) directory for the same results.
GOG, if you're listening, all users should be warned about not running as admin! Either that or change the installers' default path to something that doesn't require admin permissions to manipulate!
Good luck!