Oswald was born to Captain George Hamilton Oswald, RN and his wife Margaret Elliott Oswald (née Robertson). The Oswalds were a landed gentry family of Cavens, Dumfries and Auchincruive (now named "Oswald Hall"), South Ayrshire, Scotland, descending from merchant George Oswald, Rector of the University of Glasgow from 1797 to 1799, Oswald was educated at Beaudesert Park School and the Royal Naval College, Dartmouth. He joined the Royal Navy as a cadet on 1 May 1947. During his early career he served in the cruiser HMS "Devonshire" and was commissioned as a midshipman on 1 January 1952. He then served in the battleship HMS "Vanguard" and then the frigate HMS "Verulam". Promoted to sub-lieutenant on 1 May 1953, he was posted to the aircraft carrier in February 1955. Promoted to lieutenant on 1 June 1955, he subsequently served in the cruiser HMS "Newfoundland" and then the minesweeper .