Born Jack Sydney George Cullen in Creighton Mine, Ontario, Cullen was given the nickname of Bud by his mother when he was a young boy. He would later legally change his name to Bud. Cullen went to Creighton Mine Public School, Lansdowne Public School, and Sudbury High School before attending the University of Toronto and Osgoode Hall Law School.
In 1971, he became parliamentary secretary to the Minister of National Defence. He subsequently served as parliamentary secretary to the Energy Minister (1972) and to the Finance Minister (1974-1975).
Prime Minister Pierre Trudeau appointed Cullen to the Cabinet as Minister of National Revenue in 1975. Cullen moved to the position of Minister of Manpower and Immigration in 1976, and remained in the position when it was renamed Minister of Employment and Immigration the next year, until the defeat of the Trudeau government in the 1979 election.
Cullen lost his Sarnia seat in the 1979 election, but regained it in the subsequent 1980 election, but he did not return to Cabinet. Cullen was appointed a judge in the trial division of the Federal Court of Canada by Prime Minister John Turner in July 1984 prior to that year's election, and he remained on the bench until his retirement in August 2000.