Bickley trained at The Liverpool Theatre School. He has performed in the West End since 1981 in productions including They're Playing Our Song, The Pirates of Penzance, and as principal lead in Jukebox, Which Witch, Maddie, The Pajama Game (Best Actor nomination, Toronto), Les Misérables, Metropolis, Miss Saigon, Sunset Boulevard (playing opposite Petula Clark), and Ragtime (nominated for the Laurence Olivier and The Whatsonstage Award for Best Actor in a Musical).
Regional productions include Sleuth, No Trams to Lime Street (Best Actor nomination, Manchester), the UK première of I Love You, You're Perfect, Now Change and the world première of Black Goes With Everything. Revues include Jacques Brel is Alive and Well and Living in Paris and It's Better With a Band.
On the concert platform, Bickley has performed with orchestras throughout the UK, Europe and South America including The Royal Liverpool Philharmonic, The London Symphony, The City of London Sinfonia, The National Symphony, The Northern Sinfonia, The Royal Scottish National, The Hallé, The City of Birmingham Symphony, The RTÉ Concert, The Gothenburg Symphony, The Iceland Symphony and The São Paulo Symphony. With conductor Carl Davis, Bickley performed a series of concerts with The Royal Liverpool Philharmonic, The Hallé, The City of Birmingham Symphony, the Bergen Philharmonic in Norway and the Royal Philharmonic Orchestra.
He has appeared in concert performances of Leonard Bernstein’s Wonderful Town playing the role of The Wreck with The Rotterdam Philharmonic, and Guys and Dolls, playing Nathan Detroit with The Vienna Konzerthaus, both conducted by Wayne Marshall. Bickley has also appeared in concert arena productions of Jesus Christ Superstar and Evita.
Bickley regularly appears with the BBC Concert Orchestra. His concerts include Guys and Dolls, playing Nathan Detroit, On The Town, playing Chip, and a new musical version of Peter Pan, all at The Royal Festival Hall and recorded for broadcast on BBC Radio. Other live concert performances for BBC Radio include Bitter Sweet, Radio Rhapsody, and Big Band Special with the BBC Big Band, and many appearances as guest vocalist on BBC Radio 2’s Friday Night is Music Night. In 2000, Bickley made his début at the BBC Promenade Concerts at The Royal Albert Hall, performing in Kurt Weill’s Street Scene and a 70th birthday tribute for Stephen Sondheim.
For Grange Park Opera, Bickley has completed two seasons: Billy Crocker in Cole Porter’s Anything Goes, and Robert Baker in Leonard Bernstein’s Wonderful Town.
He was a guest artist for a gala concert in Leeds with Lesley Garrett and the Opera North Orchestra, and with orchestrator Martin Koch, he produced a concert to celebrate the work of Nelson Riddle, with The BBC Concert Orchestra and transmitted live on BBC Radio 2.
Throughout 2005–06 Bickley undertook a national tour of Cole Porter’s High Society playing the role of C.K. Dexter Haven, culminating in a season at The Shaftesbury Theatre in the West End.
His concert engagements in 2007 included Good Thing Going, a celebration of the music of Stephen Sondheim at Cadogan Hall, and The Best of Broadway, two concerts at The Royal Albert Hall, both with the Royal Philharmonic Orchestra. He also performed in several gala concerts celebrating the music of Rodgers and Hammerstein, and was a special guest for Barbara Cook's 80th birthday concert at The London Coliseum: Barbara Cook and Friends.
He is a director of The Theatre Royal Drury Lane Theatrical Fund, founded in 1766.