Okay, so here's the deal. The Wii U has 2 modes, the "Cafe OS" or Wii U portion, and the "virtual Wii"/"vWii"/Wii portion, which is exactly what it sounds like, a sandboxed Wii that works exactly like the original, thanks to some clever hardware tricks. There has been homebrew on it since it was released, using disc based exploits like Smash Stack (which I recommend). It will install the homebrew channel just like you expect, and you can install WADs and such.
HOWEVER, be careful when messing around with WADs or installing IOSes, since there is no easy/known way to restore it should you brick it. That's not to say you shouldn't still
backup your vWii just in case. Additionally, since there is now a "Boot ROM" complicating things, there will usually be a vWii specific port of certain programs, so look for it when possible.
As mentioned before, there's also the main "Cafe OS" portion, which is significantly less hacked. We have code execution and can develop very basic homebrew on versions 4.X-5.1.0
(IT DOES NOT WORK ON 5.1.1-5.2.0!). There's also a kernel exploit in the works that should allow for more advanced homebrew and a proper "homebrew channel" (although nothing will be permanent, you have to run it every time). However, since we can't run code on 5.1.1 up, we don't know for sure that the kernel exploit hasn't been patched, however given the way it works it's unlikely to have been.
That being said, if you only care about homebrew in the vWii part, feel free to update. If you care about homebrew for Cafe OS and want a safe way to update, go buy a MK8 disc, make sure your Wii U can't connect to the internet, and then update with the data on the disc. That'll bring you up to 4.1.0. However, note that you can't go online or use the eShop. If you care but still want to use the eShop and go online, us developers have said that unless you want to develop homebrew (IE using the very limited access we have with the Usermode exploit mentioned above), feel free to update. It shouldn't be that difficult to find a new browser bug and port the ROP chain, it just might take a bit, but it should definitely be possible given how old the version of WebKit they're using is.