The "Year-2038-Problem" is related to the fact that most Unix / Linux systems store timestamps as the amount of seconds since 1970. That way, a signed 32-bit value will overflow in 2038.
Both Gamecube and Wii (I'm not sure about WiiU but I suspect it'll be the same) are storing their time in seconds since 2000, so it'll overflow in 2068. Some timestamps on the Wii use a 64-bit value for milliseconds since 2000; that would overflow even later.
The Wii system menu doesn't allow you to set a date later than 2050, so you might run into trouble at that point. But Wii and Gamecube aren't really systems that rely on proper system time. If in 2050 there are any problems with the console due to the time, you could just set the date back to 2000 and live with the fact that the date the console displays on its main menu is wrong. Other than that, having a wrong date won't cause any problems.
As for games like Animal Crossing that rely on the system time, the only thing that game relies on is the fact that time is moving forward. You can set the time back a couple years and the game will continue to work.
Custom online services like Wiimmfi currently enforce that the time set on the console be correct, but that is something that is done serverside and will no longer be enforced if Wiimmfi is still around in 2050.