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Willetts is something of an enigma in that he has had no formal singing, dancing, or acting lessons. Before he was 20 he rarely visited the theatre. It was while he was working as a quality control supervisor at an engineering firm in the Midlands that he became interested in amateur dramatics. He came to the attention of Bob Hamlyn, artistic director of the Belgrade Theatre, in Coventry after his sensitive and emotional portrayal of Charlie Gordon in the Charles Strouse musical, Flowers for Algernon. Hamlyn cast him as "third flunky from the left" in another show by Strouse, Annie. It was at this time, while Willetts was in his thirties, that he began his meteoric rise to the top.
Director Trevor Nunn put Willetts into the chorus of the original London production of Les Misérables and within a year he was understudy to Colm Wilkinson in the lead role of Jean Valjean, which Willetts eventually took over when Wilkinson left in 1986 to join the Broadway company. In 1987, when Michael Crawford departed London for the US premier of Andrew Lloyd Webber's The Phantom of the Opera, Willetts succeeded him at Her Majesty's Theatre in the West End's hottest ticket. He subsequently played the role of the Phantom in Manchester on the UK tour to critical acclaim, winning an Evening News Theatre Award. In 1990, he was given his first opportunity to originate a role when he appeared opposite Petula Clark in Someone Like You, a musical for which she had composed the score.
Since then, Willetts has appeared in a concert version of Jesus Christ Superstar, taken the lead role in Leicester Haymarket's acclaimed production of Stephen Sondheim's Sweeney Todd, appeared as Old Deuteronomy in the 20th anniversary production of Andrew Lloyd Webber's Cats in London, and as Jean Valjean in the 10th anniversary production of Les Misérables in Sydney, Australia. He created the role of Heathcliff in the studio recording of Bernard J. Taylor musical version of Wuthering Heights.
In 2004 he appeared in the West End production of Ragtime alongside Maria Friedman. In December of the same year he appeared as Captain Hook in Peter Pan in Bromley. Most recently he has played the starring role in Seven Brides for Seven Brothers in the West End (2006). He recently played the role of Emile de Becque in a UK touring production of Rogers and Hammerstein's South Pacific, which toured the UK until July 2008.
Dave is currently reprising the role of Julian Marsh in a UK Tour of 42nd Street for UK Productions. Dave appeared in this production in 2007 as a replacement for Paul Nicholas. For the 2012 production he is joined by fellow stage veteran Marti Webb.