Considering the latest leaks and rumours about the NX, it seems that it will be some sort of tablet that docks to the TV to work as a hybrid of sorts. That makes me wonder how would Nintendo use their many add-ons/peripherals on the console. The Wiimotes, gamepads and Pro Controllers would be a safe bet, but I doubt others like the Wii Fit Board or the GC Adapter would work on that setup (shame, I have two GC Adapters, one from Nintendo and one 3rd party).

Whichperipherals/add-ons/controllers will work on the NX?

Hopefully none. Nintendo just needs a clean break from the Wii U. Just forget it existed and move forward.

I'd be delighted if they added gamepad and pro controller support. Otherwise I want nothing from the Wii/GCN/Wii U era.

Maybe the pro controller. Outside that one, i think none. Wii era is ending for Nintendo so likely the controls it brought also.

GCN probably, lol.

What can work?
Anything that is connected wirelessly: bluetooth (pro, wiimote, wiifit board) and 802.11n (gamepad).
Anything with USB Type-A: hdd, microphone, lan and gamecube adapters; but maybe they'll go Type-C.

What will work?
No clue, for the controllers my guess would be wifi 802.11n, but they might stick with bluetooth.
Cable connectors I'd also guess they'd stick with USB Type-A 3.0, but it would be cool if they used Thunderbolt 3.

I'm happy enough just with GC controllers and Pro controllers. 

I hope they ditch Wii era stuff entirely. I hate the design of the Wiimotes and Gamepad -the sooner theyre gone the better, imo. They should not rely on 2 gen old controllers to carry their system

I think it will be all-new controllers. But the detachable controls on the tablet will function as Wii Remotes for backward emulation.

I feel that it'll be all new, but the only thing that'd make sense to me that would work still is the Wii U pro controller for when playing the NX on your TV. But that's also if Nintendo choose to keep the same format of controller/buttons for the NX (ie. no Touch pad/screen required etc..)