well they have said they intend to use the Wii U's architecture going forward so I'm assuming that Wii U games will play on the next console, so they wouldn't necessarily need to go dark for a significant amount of time. hell they could even boast about early 2016 Wii U games at the release of their next console as pseudo launch titles.
I think it really depends how much they plan to unify handheld and console software/hardware. They need to be working on a launch lineup for the 3DS' successor anyway so if they're both handheld/console games then it's definitely feasible to have the software to back up a new launch.
Personally I didn't read that statement as "The next handheld will be able to play Wii U games." because from a technical perspective, that's highly improbable, but rather that the hardware feature set of their next handheld will support all the hardware features found on the Wii U and more.
Like if the 4DS uses a modern SoC, it will support DX11.2 and OpenGL ES 3.0 hardware features, which means they can take their Wii U engines and port them to the handheld without huge problems. Similarly, the next console would assuredly support the same hardware features, allowing the engines used on 4DS to be used on that as well.
Currently the 3DS is an OpenGL ES 1.1 device (no programmable shaders, lots of other limitations) whereas the Wii U is a DirectX 10 tier device (programmable shaders under shader model 4, notably more flexible, but not quite as flexible as DirectX 11 tier hardware). Due to this 3DS engine technoloyg is woefully insufficient for the Wii U while Wii U technology is functionally incompatible with the 3DS.
By having the 4DS and the Wii 3 share the same hardware feature set, even if they have different hardware power, they can share the same internal engines and to an extent the same art assets (the 4DS versions would have to have lowered poly counts, but there's lots of programs to help with that) in the same way that Unreal Engine 4 can run on a modern iPhone and the PS4 or Plants vs. Zombies: Garden Warfare can run on iOS8 devices with near console quality despite being on Frostbite 3.
Basically if they use the same hardware feature set they can take their Wii U technology and upgrade it to modern standards once, and then use that as the base for developing both 4DS and Wii 3 games, and even have the option of running them on both systems if they pare down the 4DS versions visually in a relatively automated way.