As others have noted, Nintendo had significant problems with HD development with the Wii U which they would have had with a 2006 version as well. So games in general would have taken longer to make. There;d be no Wii Sports, Wii Fit, or Wii Play. Twilight Princess would be a Gamecube exclusive.
I'm not even sure the system would get that many more 3rd party multiplats. Mass Effect 1 and Bioshock 1 were Xbox exclusives at first. Assassin's Creed was supposed to be a PS3 exclusive went multiplat because Ubisoft saw the lead the 360 had built in its first year on the market alone. This hypothetical Nintendo system wouldn't be successful enough to warrant bringing that over. It would probably get Japanese games like Final Fantasy 13 and Resident Evil 5, but Japanese developers like Square and Capcom were not at the top of their game that gen anyway. Eventually developers might port games over if it's easy and cheap, but they won't be high-effort, high-quality ports, but cash grabs. The 360 versions will be the best.
Literally everything that created hype for the Wii and sustained the system's momentum in the first 4 years would be gone, the first Mario would probably come out in 2008, the first Zelda probably in 2011 like Skyward Sword. Nintendo would need to put out shovelware just to have a lineup in 2006-2007. Best case scenario it sells N64 numbers based on the market growing. Worst case scenario it sells Wii U numbers because of no momentum, no killer app, and no library for 1-2 years.