As it happens, my Wii U, hooked up to the TV with an HDMI cable, suffers from a noticeable split-second lag. Whether this is the fault of the Wii U itself, of the use of an HDMI cable, of the TV, or of how I've got it set up, I don't know. But I haven't been able to completely fix it. Most games are still playable, though a little harder (I kept missing platform-edge jumps in NSMBU). Rhythm Heaven Fever, however, is totally unplayable.
So I've got the Wii hooked up to the TV alongside the Wii U. Pretty much just for that game.