I pledge to fight this “noughties” thing. I know they are using it on TV retrospectives and what have you, but I have never heard anyone say it in real life, and I reckon that is where decades get their nicknames. I have always said that the decade just finished will be known as “the two thousands” (because IME people say “two thousand and nine” etc., not “twenty-oh-nine” and certainly not anything involving the word “nought”). I don’t believe “noughties” will stick, but it’s certainly the leading contender among smart-arse names for the decade.
As for this decade, I think we will tire of saying things like “two thousand and fourteen” so it will be “twenty fourteen”. So I would guess that the the decade will be the “teens”, regardless of the fact that not all of the years have “teen” in them.