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.