When you launch the first download, the gog downloader will ask you where to put downloaded files. The default location is "My Documents" folder.
If you have trouble finding downloaded file, you can always do a search for a file named "falllout". Just click Start -> Search -> Search for files and folders (on Win XP).

You can try to uninstall the downloader, and then to install it again. This should fix the missing window. But unfortunately it won't help you to find those missing files. And if your OS can't find them, it looks like you'll have to download them again.