There was a list online of all the awards but I can't seem to find it. So this list is likely incomplete:
Hugo Awards - won 3 consecutive years, all for Steven Moffat scripts, and beating Battlestar Galactica almost every time.
Saturn Awards - won Best International Series in 2008.
Constellation Awards - It won 3 awards in 2007 - Best Actor (Tennant), best SF episode for Girl in the Fireplace, and Best SF Television series. It also won Outstanding Canadian Contribution to SF due to the CBC network contributing seed money to the show, which made it eligible for that category.
BAFTA TV Awards: Audience Award and Best Drama Series in 2006; Best Writer (Moffat) in 2008; Best Editing (2009), and an armful of nominations
BAFTA Cymru Awards (a BAFTA spinoff focusing on Wales-based productions): It's won 10 of these so far, including a whack of technical awards in 2008.
National Television Awards: It's won Most Popular Drama every year since 2005; Eccleston won Most Popular Actor and Tennant has been unbeatable since 2006; Billie Piper won Most Popular Actress twice. Both Tate and Agyeman were nominated as well.
Royal Television Society - won for Best Sound in 2008
People's Choice Awards - yes, the American awards. It didn't win but it was nominated along with BSG in the Favorite Sci-Fi Show category. Stargate Atlantis won in what is generally assumed to have been a split vote between DW and BSG.
TV Quick Awards (never heard of this one either) - DW has won an armful of these since 2006.
Visual Effects Society Awards, 2008
Writer's Guild of Great Britain Awards, 2007 - the entire writing team was honored.
I'm certain there have been others. A few of the above like the NTAs were viewer-voted awards, indicating the popularity of the show over there, while the BAFTAs are juried. People can bitch all they like about the new show and RTD, but the fact is while the original series picked up the occasional award (The Writer's Guild of Great Britain gave its writing team the same award as above in 1975; the Royal Television Society gave it Best Graphics in 1974; TV60 named it Most Popular Drama Series in 1996 (tribute award), and it got a couple of BAFTA TV nominations in the mid-70s), the new version has been raking them in. And considering this has been the era of Rome, The Tudors, Life on Mars, Being Human, Hustle and scores of other acclaimed and, arguably, more internationally popular series, DW's success really has been saying something.
I particularly like to point out nuWho's consistently high Appreciation Index ratings and its NTS success to people who make the claim that "nobody likes it". And I still see that chestnut rolled out by the occasional particularly disillusioned fan who longs for the days of papier mache spaceships and John Nathan-Turner.
Alex