Update: Nope, it just works with Smash Bros.
http://www.nintendoworldreport.com/news/38673/gamecube-controller-adapter-only-compatible-with-smash-bros-on-wii-u
Update: Nintendo confirmed that the GameCube Controller Adapter will only work with Super Smash Bros. on Wii U. The original story is still below, even if it is just kind of a bummer now.
The upcoming GameCube Controller Adapter will be backwards compatible with all games that support the Wii U Pro Controller and Classic Controller.
According to the description on Nintendo's UK online store, not only will the adapter be compatible with Pro/Classic controller games, but also games through the Wii Mode on Wii U, as well as the Wii itself, if you happen to own a Wii without the GameCube ports.
Nintendo announced the Wii U GameCube Controller Adapter a few weeks before E3 as a means for hardcore Super Smash Bros. players to play the upcoming Wii U release to play with their preferred controller, but this is the first confirmation that it's compatible with something other than Smash Bros.