However - now that we have a historical perspective, what is the answer to the OP?
In the UK at least “Noughties” seems to be the default, at least for the radio presenters that I listen to.
Now that we have the historical perspective that was lacking in 1999, I suppose this is a good enough reason to bring back this zombie thread. As for the name for the decade 2000-2009, I have heard “the two thousands”. Which is not entirely unambiguous since it could also be interpreted as referring to the entire century or even the millennium, but in the instances in which I have heard it, it was clear that the years 2000-2009 were meant.
I think the definitive answer has to come from the cable channel VH1, which ran a nostalgia series called I Love the '80s. They followed that with additional series called I Love the '70s, I Love the '90s and I Love the 2000s.
Or another way to think of it; what do high school and college graduating classes call their years? I graduated high school in 1984 and we called ourselves the class of eighty-four. Did the high school graduating class of 2002 call themselves the class of two? Class of Oh Two?