I was browsing the steam forums on this game, from 2014 and people were having the same issue I am having right now.
You cannot select gamepad from the options menu.
It just won't let you turn on gamepad. It's unselectable.

Please check with the developers on this game and ensure GOG has the most up to date version.

I cannot enable gamepad.
This is a serious problem.

I don't have a gamepad myself but I gifted the game to a friend who has been playing it with a PS3 controller with no issues whatsoever. What controller are you trying to use to play the game? And, since we're at it, what are your overall specs and OS? That might help narrow down your particular issue with the game.

Also, I don't know if you tried this already or not, but the G key toggles gamepad on/off, you might want to give that a go, if you haven't already.

Unfortunately, I'm not tech savvy enough to help you with this, but hopefully the info you provided will be enough for someone else more in the know to help you sort out your issue.

The build of Nuclear Throne GOG has right now doesn't support daily missions and challenges, but those will come eventually, so, if it is, in fact, a problem with the game, I'm pretty sure it'll get fixed any time soon.

I don't think it's an issue with your OS, since Windows 7 is a supported operative system, everything should be fine on that end. As for the controller, I really don't know. I can safely say the GOG build works with a PS3 controller with no issues whatsoever, so maybe it's a controller thing; perhaps toggling the gamepad on via menu isn't working for you because the game doesn't recognize your Logitech as a valid controller. Like I said, try using the hotkey instead of the menu; according to the official wiki for Nuclear Throne, maintained by Vlambeer themselves, G should be the key to toggle gamepad on and off without having to go into the options menu. Maybe that works for you?

I would wait for a reply by someone other than me, hopefully they'll be able to address this specific problem and try to solve it for you. Another option is to submit a trouble ticket to GOG Support, they're definitely more in the know than us forum users.

If I hit 'g' to swap to gamepad mode, that 'works' insofar as setting the game up to read a controller.

But the game doesn't read a logitec controller. That's probably the main issue.

I wonder why the developers would not support logitec and only those system controllers.

I don't own any consoles and am not going to go out and buy a console controller just for this sort of thing, when my logitec has worked nicely for all modern games.

Okay so digging deeper, I find that this game is on three platforms:

PC
Playstation
PS Vita.

It also seems to have been slightly rushed out in the PC port, and now updates are slowly creeping forward.

Because of the playstation platforms, it has playstation controller support built into it, but with the rushed PC port, logitec support wasn't added.

It 'seems' there are updates and patches coming (unrelated to the controller support but I imagine logitec support will be among the updates).

I really wish I had done more research on the game before buying. I don't want a refund or anything, but I would have waited until a more patched PC version. It 'is' clearly a rush port.