Notice that we did not find the CD key here? That's because the key is not stored in registry, but in one of the packed files (encrypted too, if I remember correctly). Play the game, grow tired of it, come back to NS. Load those two registry files into registry by doubleclicking on them. Transfer the two registry files saved in the steps above. Transfer ALL your War3 files from the old computer to the exact same location on the new computer (This is important if you want that automatic update to operate without problem). (Do this for both HKEY_CURRENT_USER\Software\Blizzard\WarCraft III and HKEY_LOCAL_MACHINE\Software\Blizzard\WarCraft III) Select tht WarCraft III key, choose Registry -> Export in the menu, and choose a name for the file.
Find your way to the War3 keys (probably under HKEY_CURRENT_USER\Software\Blizzard\WarCraft III and/or HKEY_LOCAL_MACHINE\Software\Blizzard\WarCraft III (I'm not sure of the exact location here, don't got WC3 installed on this computer)). Open regedit (Start -> Run -> type 'regedit')