I just bought and installed the game. I'm stuck on the main menu. There's no mouse cursor and it's not getting any of my keyboard inputs either, with the exception of Alt+F4 to close the game.

Because of this problem I can't do absolutely anything. Happens on full screen and both windowed modes.

I also disabled the GOG Galaxy overlay and the problem still occurs.

"Literally unplayable" has been nothing but a meme in my eyes, until now...

Is this running within galaxy or with out?

With Galaxy.

I also just discovered the solution. The game defaults to using the controller if it detects any, and completely locks out the keyboard and mouse inputs.

I unplugged my controller and that was it.

Yeah - The same happened with the Steam version, so I was not surprised. I always play FPS games with M+KB and other games like JRPG games with a controller. So I always have a controller plugged in. By the way - Fallout 4 is not the only game that works that way, so...

As an alternative to unplugging your controller, maybe this would work:
https://www.pcgamingwiki.com/wiki/Fallout_4#Mouse_cursor_not_visible_in_menus

There is also even a setting in the Gameplay section to turn the controller off entirely. You'll probably want to do that, then it's never a problem.

Thanks a lot for that...
bGamepadEnable=0

was the solution.
How the hell am i supposed to play Fallout with my Steering wheel?