Established in 1953, the Hugo Awards are awarded to the best science fiction or fantasy works and achievements of the previous year, as voted for by members of the World Science Fiction Society. The awards themselves are presented at the annual World Science Fiction Convention over a number of various categories.
The Doctor Who Universe has had the most success in the category of "Best Dramatic Presentation, Short Form", which is awarded to "a dramatised production in any medium", which generally lasts less than 90 minutes. 
Doctor Who itself has been nominated every year since its revival, including multiple nominations from 2006 to 2014, with a total of six wins. Torchwood also received a nomination for Captain Jack Harkness, and the 50th Anniversary stories An Adventure in Space and Time and The Five(ish) Doctors Reboot were both nominated in 2014.
Steven Moffat in particular has had unparalleled success in this category. As an individual writer he has been short-listed more than fifteen times. Of these, he received two nominations in both 2011 and 2014, and in 2013 he had written three of the five nominees. He has also won the award four times, including three consecutive wins for his first three televised stories.
The only other category in which the Doctor Who Universe has won a Hugo Award is for the "Best Related Work", which is awarded to "work related to the field of science fiction, fantasy, or fandom".  This was for the reference book Chicks Dig Time Lords in 2011.
The 2019 award was won by Archive of Our Own, "A fan-created, fan-run, non-profit, non-commercial archive for transformative fanworks, like fanfiction, fanart, fan videos, and podfic", of which the Doctor Who universe (spread out over various categories) is one of the most represented fandoms.
Paul Cornell and Jimmy Broxton's The Girl Who Loved Doctor Who was nominated for a 2014 award in the "Best Graphic Story" category, which is given to a "science fiction or fantasy story told in graphic form".
This makes Cornell the only writer to earn Doctor Who nominations in two separate categories. He previously wrote Father's Day and Human Nature/The Family of Blood, which received nominations for the best Dramatic Presentation, Short Form in 2006 and 2008 respectively.
Non DWU winners
As well as the award winning episode The Doctor's Wife, Neil Gaiman's writing has also achieved five other Hugo Awards, as well as a further nomination. These have all been awarded over five separate categories: "Best Short Story", "Best Novel", "Best Novella", "Best Related Work", and "Best Graphic Story". The 2008 award for "Best Dramatic Presentation, Long Form" was also won by the film Stardust, which was adapted from his novel of the same name.
Other contributors who have won a Hugo Award for non-DWU works include Harlan Ellison (multiple, 1966-86), Jane Espenson (Dramatic Presentation, Short Form, 2003), and Alan Moore & Dave Gibbons (Other forms, 1988). Nominees include Douglas Adams (Dramatic Presentation, 1979), and Michael Moorcock (Professional Magazine, 1968-69)