Unicomb began acting as a child with ABC Radio's
Children's Session in 1936. In 1951, Unicomb had a small part in the film
Captain Thunderbolt, played Oswald in a well-reviewed
King Lear for
John Alden and, famously, an athletic Hamlet in
Robin Lovejoy's production at Sydney's
Metropolitan Theatre. On the strength of his performance in
Hamlet, Unicomb won a place in
Ngaio Marsh's company, touring New Zealand. Unicomb relocated to the United Kingdom, to further his acting career. Around 1956, he appeared (as ‘father’) alongside wife
Fernande Glyn (as 'mother') in Dennis Driscoll's
Off the Deep End, for the Colwyn Bay Repertory Theatre in Wales. In 1958, the pair returned to Australia, where they appeared in several productions together. In mid-1959, Unicomb appeared with Glyn as part of
J. C. Williamson's Shakespeare Company in a Melbourne production of
The Merchant of Venice in Melbourne. The following year, they appeared in a regular radio comedy,
Mr. and Mrs. – as part of
Omnibus on Sydney's
2GB. In late 1962, both Glyn and Unicomb had lead roles together, alongside
David Hutcheson and Martine Messager in another
J. C. Williamson’s production, a play adapted from
Marcel Achard's ''
L'Idiote, called A Shot in the Dark
, a French comedy which ran for nearly four months in Sydney and Melbourne. Around 1965, Unicomb and Glyn were guest stars together on The Mavis Bramston Show''. Unicomb later worked as a comedian. An audio recording of Unicomb's theatre recollections is held by the
National Library of Australia in Canberra. ==Personal life==