Alldritt was educated at St Francis de Sales Catholic Junior School, and the University of Liverpool.
He served in the Armed Forces during World War II, then became a trade union official.
He also served as president of the Liverpool Labour Party and Trades Council, at the time, the youngest person to have held the post.
[1] He was elected as Member of Parliament (MP) for Liverpool Scotland at a by-election in 1964, and held the seat until his resignation from the House of Commons in 1971.
[1] In 1970, Alldritt was appointed as regional secretary of the National Union of General and Municipal Workers for Liverpool, North Wales and Northern Ireland, serving until 1981.