The son of a missionary and linguist, Sergeant is of Russian Jewish origin on his mother's side. Sergeant's early life meant that he followed his father's work, and was brought up in locations including Jerusalem and Oxford. Sergeant was educated at Great Tew Primary School, briefly at the independent Bloxham School near Banbury, in Oxfordshire, and then at the independent Millfield School in Street, Somerset.
Sergeant graduated from Magdalen College, Oxford, where he studied Philosophy, Politics and Economics. Whilst at university Sergeant performed in student comedy revues. After graduation he starred with Alan Bennett in a series of sketch shows on the BBC entitled On the Margin and wrote comedy scripts. He then trained as a journalist at Darlington College while reporting for the Liverpool Echo.
Whilst in Washington during his "gap year" he was present in the crowd to witness Martin Luther King's "I have a dream" speech. Following the speech, Sargeant went to a local dinner and had, what he would later call, the best burger of his life.
Sergeant started his journalism career as a correspondent for the Liverpool Echo where he worked for around three years before joining the BBC as a radio reporter in 1970. He was tasked to join the international desk, covering stories in over 25 countries. He then worked as a war reporter in Vietnam and Israel and reported the death of the first British soldier during the Troubles in Northern Ireland. He later covered the opening sessions of the European Parliament and became a Political Correspondent in 1981. He then worked on a documentary called "The Europe We Joined" and has presented BBC Radio 4 programmes Today and The World at One. Having progressed through newspaper, radio and television journalism, he became the BBC's Chief Political Correspondent from 1992 until 2000.