There's the now-abandoned Tri-Core Linux on the vWii side; though nothing in Wii U mode as far as I know. Linux would be doable, though it's a huge amount of work for anyone who's not experienced in that sort of thing. If it were a thing, it wouldn't actually be that barren in terms of programs - PowerPC was the architecture used in Macs until the mid 2000's. Since they're, y'know, Apple products, Debian supported the platform for quite a while (only dropping the 32-bit variants recently). Don't expect Steam, but you can expect most established, older open-source products to be written with PowerPC support somewhere, though you may have to compile them yourself.
Back to the main question - Linux is totally possible, but it won't happen anytime soon (if at all). You also wouldn't be running any x86 software outside of a virtual machine.