My installation is fine and dandy, but for some reasons my mods won't work while installing them with Nexus Mod Manager. It says they are activated, and they have a check mark, but no mods seem to work when I load up TW3.

Looking around on the forums people have said something about this being in the first directory Local Disc, so I have been trying to move it to Program Files x86. It move the folder, and uninstall the game, but when trying to reinstall it in the new location (I changed install location in the settings) it'll check the files, but then will tell me that I do not have enough disc space. I have 860GB. If I move it back to its original location, it installs fine.

Any help? Is there another way I can get the mods working?

I wouldn't install any games into Program Files since Windows' security features are messing with this folder. Most of the time it's better to put them into a separate folder on C:\
If you get an out of disk space error, this could be because of temporary files during installation. Depending on where the installer or Galaxy extracts the files on your system, this harddrive could run out of space, no matter where the final installation will be.

Program Files is evil. You can't modify anything in there without Admin rights.

Yeah, as DeMignon says, installing into Program Files is generally not a great idea. If you're using a mod manager, you could have a modified .exe that you need to run. I haven't used mods for Witcher 3, but for, say, Bethesda's games, once I've installed the mods, I have a different .exe to run to make sure everything works and plays nice. Could that be the case for you?