I have bought a few third party Wii controllers over the years. My experience is they often don't work as well as the genuine ones. But as the the genuine ones age they sometimes aren't perfect either. Wii controllers have a lot of sensors and with third party and/or old ones there always seems to be something that doesn't work quite right (buttons, motion sensors, especially in the nunchuk, battery indicator, infrared). So I would definitely go for genuine ones if you can, but I know it's a lot of money.
The same controllers work on Wii and Wii U. They're bluetooth. You should know that some games require "MotionPlus/Wiimote Plus) but most games don't. The Motion Plus controllers have additional sensors for fine motions. Some games that require motionplus include Skyward Sword, Wii Sports Resort.
There's a NZ Nintendo Facebook group by the way where a lot of folk hang out. It's a lot of Pokemon stuff but other stuff as well.