HTH!

Over the first decade of this century (or of the 21st century) is a possible alternative

This has the benefit of brevity, and avoids the whole question of what to call this decade. I don't think it's necessary to over-translate here.

this is what came to mind before I looked at the other suggestion

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Note added at 2 hrs (2013-05-23 12:40:56 GMT)
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for suggestion read suggestions !

Depuis le début des années 2000 = from/as of 2000.

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Reading it is another thing!

You have to translate "depuis le début des années 2000". Since "the noughties" are deemed unacceptable for the register of your source text, this could be the alternative.

on the theory of 'the more the merrier', here's my take

Schools in deprived areas stand to lose most in budget shakeup ...
www.guardian.co.uk › Education › Schools‎
Nov 18, 2011 – Little has changed in the way schools are funded since early in the new millennium. The researchers said there was wide variation between ...

Africa investment 'will rebound' - SouthAfrica.info
www.southafrica.info/pls/procs/iac.page?...‎
Feb 17, 2009 – "Domestic reforms have fostered Africa's robust growth since early in the new millennium," the research report states. "Widespread economic ...

The BBC at least has stopped saying 'two thousand and six' and started saying 'twenty-twelve'

Just as we once said 'the nineteen-hundreds', I have heard some people (inclduing the BBC) using this expression; of course it could refer to the entire 21C, but in practice, I don't think one would ever make that mistake — unless, of course, it was "in the mid-twenty-hundreds" etc.

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'since the early twenty-hundreds' would probably be fine, as indeed would 'since the very start of the 21st century'

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I don't think 'since' would in fact be right here — maybe simply 'from'? Or 'right from' if there is a need to emphasize the earliness...

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From the way it is being used here, I really don't think this is talking any differently from if we'd said 'the eighteen hundreds' or 'nineteen hundreds' — just the early part of the century.

I don't see anything in the source text to say that there is a specific connection with its being a whole new millennium — which is just something everyone got a bit obsessed about when the Year 2000 came round; from today's perspective, it's just yet another century like all the others — unless, of course, one is specifically referring to it in the wider perspective of a whole millennium; which I don't honestly believe is the intention here.

As written. Completely consistent with the 1960s, 1970s, 1980s, and 1990s.

How you *say* it is up to you.:-)

I believe

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http://en.wikipedia.org/wiki/2000s_(decade)

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or the 00's/zeros/2000's

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In the English-speaking world, a name for the decade was never universally accepted in the same manner as for decades such as the '90s, the '80s, etc.[10][11][12]
Orthographically, the decade can be written as the "2000s" or the "'00s". Some people read "2000s" as "two-thousands", and thus simply refer to the decade as the "Two-thousands" or the "Twenty-ohs". Some read it as the "00s" (pronounced "Ohs", "Oh Ohs", "Double Ohs" or "Ooze"), while others referred to it as the "Zeros".[11][13] The single years within the decade are usually referred to as starting with an "Oh", such as "Oh-Seven" to refer to the year 2007. On January 1, 2000, the BBC listed the noughties (derived from "nought"[14] a word used for zero in many English-speaking countries), as a potential moniker for the new decade.[15] This has become a common name for the decade in the UK[16][17][18][19][20] and Australia,[21][22] as well as other Anglospheric countries.
Others have advocated the term "the aughts", a term widely used at the beginning of the twentieth century for its first decade.[23][24] The American Dialect Society holds a lighthearted annual poll for word of the year and related subcategories. For 2009, the winner in the "least likely to succeed" category was "Any name of the decade 2000–2009, such as: Naughties, Aughties, Oughties, etc."

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typo above: should be '00s of course

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http://www.guardian.co.uk/lifeandstyle/gallery/2009/oct/16/t...

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http://moreintelligentlife.com/content/nick-coleman/defining...

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http://www.academia.edu/430295/Ancillary_Relief_in_Northern_...

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"the 2000s decade"


is used here
http://www.innovatemyschool.com/industry-expert-articles/ite...


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http://www.macmillandictionary.com/buzzword/entries/noughtie...
....
The interesting thing about all this is that it's only relatively recently, certainly at the back end of the decade, that we finally decided to plump for the noughties as the recognised English term for this period of time.

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Note added at 12 hrs (2013-05-23 22:42:56 GMT)
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Could I just remind people that the ORIGINAL QUESTION was asking for the equivalent of

"les années 2000 (and this will be for Glossary as well)

"si vous avez : les années 60, vous direz les "sixties", les années 80 : "the eighties" etc"

I gave several possible answers for this. My preferred option and one I believe to be used in many contexts, with nothing humourous or frivolous intended is the NOUGHTIES. Just look at the links or Google for yourselves.
I already gave the macmillan link. here it is in the OED and Cambridge

http://oxforddictionaries.com/definition/english/noughties

http://dictionary.cambridge.org/dictionary/british/noughties


Anyway, I also offered

the '00s/zeros/2000s as alternatives.

The question was NOT about translating the ST so it's a bit of a wild goose chase imo to translate

"Depuis le début des années 2000" which is not the headline term


As for what the period 1900-1909 was called, I believe it was most commonly called "The Edwardian era" since that spanned pretty much the same period of time.

One of the terms that has been used.

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Note added at 1 day1 hr (2013-05-24 12:06:42 GMT)
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Rereading the question, I think that, seeing the seriousness of the subject involved, not one of the cutsey names for the decade is appropriate, something along the lines of "the first years of the XXIst century" would work much better.

I agreed with "Noughties" before - but, strike that! Noughties is too informal for this text.