Bleakley works as an electrician in the Harland and Wolff dockyards while becoming increasingly active in his trade union. He studies economics at Ruskin College in Oxford, where he strikes up a friendship with C. S. Lewis, about whom he later writes a centenary memoir. He later attends Queen’s University Belfast. A committed Christian, he is a lifelong Anglican – a member of the Church of Ireland. Throughout his life, he is a lay preacher.
Bleakley joins the Northern Ireland Labour Party (NILP) and contests the Northern Ireland Parliament seat of Belfast Victoria in 1949 and 1953 before finally winning it in 1958. At Stormont, he is made the Chairman of the Public Accounts Committee, but he loses his seat in 1965. He is head of the department of economics and political studies at Methodist College Belfast from 1969 to 1979.
Bleakley runs for the Westminster seat of Belfast East in 1970 (gaining 41% of the vote), February 1974 and October 1974 for the Northern Ireland Labour Party each time, but never enough to win the Westminister seat from the Ulster Unionist Party (UUP). In 1971, Brian Faulkner appoints him as his Minister for Community Relations at Stormont, but as Bleakley is not an MP, he can only hold the post for six months. He resigns five days before his term expires in order to highlight his disagreement with government policy, specifically the failure to widen the government to include non-Unionist parties, and the decision to introduce internment. He writes a respectful biography of Faulkner and his own memoir of the period.
After the Parliament is abolished, Bleakley stands for, and is elected to, the Northern Ireland Assembly and its successor, the Northern Ireland Constitutional Convention. He stands again for Belfast East in the February and October UK general elections, but wins only 14% of the vote each time.
By the late 1970s, the NILP is in disarray, and does not stand a candidate for the 1979 European Assembly election. Bleakley instead stands as an “Independent Community Candidate,” but takes only 1.6% of the votes cast.
During the 1980s, Bleakley sits as a non-partisan member of various quangos. From 1980 to 1992 he is general secretary of the Irish Council of Churches. In 1992, he joins the Alliance Party of Northern Ireland and is an advisor to the group during the all-party talks. For the 1996 Northern Ireland Forum election, he is a prominent member of the Democratic Partnership list and stands in Belfast East, but is not elected. In 1998, he joins the Labour Party of Northern Ireland and stands in Belfast East in the Assembly elections, receiving 369 first preference votes.
Bleakley dies in Belfast at the age of 92 on June 26, 2017.