David won two awards for his portrayal of Hamlet and was nominated for several. He missed out being eligible for even being nominated for the prestigious Olivier Award due to his back injury during the West End transfer of the show. The Olivier is given only for London performances of a play and David was not on stage in London enough days to qualify. It was widely speculated that he would have won if he had been nominated. Here is a quick video of Patrick talking about David's talent:
He did win the 2008 Critics' Circle John and Wendy Trewin Award for Best Shakespearean Performance, which was awarded jointly to David for Hamlet and to Sir Derek Jacobi for his performance as Malvolio in Twelfth Night.
From The Telegraph - January 27, 2009 - Stephen Adams
"In accepting his award Tennant, 37, said: "When I was at drama school, I waited at the stage door of Glasgow Theatre Royal to get Derek Jacobi's autograph, after his breathtaking Richard II.
"I was utterly inspired by that night, so to be sharing a prize for
Shakespeare with such an amazing actor means more than I can say."
He also described the experience of playing Hamlet for the Royal Shakespeare
Company at Stratford-upon-Avon as "a dream I never imagined I would get
to fulfil"."
He was also awarded the 2009 Theatregoers' Choice Awards for Theatre Event Of The Year: David Tennant's return to the stage as Hamlet. Over 30,000 people voted online for the awards and he won by 53% of the vote in this category.
He was unable to attend but Ed Bennett and Gregory Doran
accepted the award for him. “He couldn't be here himself because I've
kicked him in the back again” joked Bennett. Later on in the press room
Ed Bennett said "If there was an award for most scared understudy I
think I would have been up there among the ranks but as it is I'll
collect this and maybe give it to him if I ever see him again!"