Has to be said, I've never come across a term for the noughties sorry. Guess we never use one.
Update: I've since come across y dimdegau once or twice for "the noughties". It's totally intuitive and would be easily understood by Welsh speakers who'd never heard it before. I like it!
I've not seen any common use word in English either. I've heard "the noughties" but it isn't popular
"Mileniwn", I think. (Trying hard to remember the spelling of the song title)
I personally prefer dwyfilau for 'two-thousands' (assuming that's how you'd represent such a thing)