He did radio acting work with Hugh Denison's BSA Players (for Broadcasting Service Association, later to become Macquarie Players). He came to the attention of Australian Broadcasting Commission radio drama producer Lawrence H. Cecil, who was to act as his coach and mentor throughout 1939 and 1940. He was "Chris" in the Children's Session and the first Muddle-Headed Wombat.
Finch co-wrote and directed an award-winning short film, The Day (1960) and announced plans to direct a feature but it did not eventuate. He won his third BAFTA for Best Actor for No Love for Johnnie (1961), although like Oscar Wilde, the film lost money. He was originally chosen to play Julius Caesar in Cleopatra (1963) and filmed scenes in London, but when the film was postponed he withdrew; the role was recast with Rex Harrison.