You could export your nand to an external device and use neek. But the experience of running neek will be a bit different than using your real nands file system.
There is a tool called punetwiin (I hope I spelled it right), similar to its predecessor Betwiin, than can convert a nand dump to be compatible with a different wii provided you feed it both sets of wii keys. It's not really recommended because if u don't have bootmii boot2 or a hardware flasher the risk of brick is more than most ppl would be comfortable with. So while technically it can be done I would advise against it.
Ultimately, I think the best solution would be to use modmii (or similar reputable guide like wii.guide) to efficiently modify the new console, then just install the extra channels/wads you want, maybe some miis and gamesaves using save game manager gx, and you're all set. And it would probably be faster than working out the complexities of converting a donor nand to a recipient console using punetwiin.