Osborne was born in Orange, New South Wales, and educated at North Sydney High School and Sydney Church of England Grammar School. He graduated with a degree in law from the University of Sydney. He joined the Royal Australian Naval Volunteer Reserve in 1938, and with the outbreak of the Second World War, he was seconded to the Royal Navy in 1940. He was awarded a Distinguished Service Cross in 1940 for "bravery and devotion to duty" while assisting the evacuation of forces from Norway as a sub-lieutenant on the "St Loman", an armed trawler. He then successively commanded , and , escorting ships between the United States and Canada and the United Kingdom in the Battle of the Atlantic. He crossed the Atlantic 22 times and was the only Royal Australian Naval Volunteer Reserve officer to rise to the command of a Royal Navy destroyer during the war. In 1945 a Bar was added to his DSC for sinking of a German U-boat.