On 15 May 1952, Doggett was made prefect of the newly erected Prefecture Apostolic of Aitape. On 11 November 1956 when it was elevated to a Vicariate Apostolic, he was appointed its vicar apostolic and Titular Bishop of
Mundinitza. He was consecrated as a bishop on 26 February 1957 at
Mary Immaculate Catholic Church, Waverley by
Romolo Carboni. In 1968, he helped to bring the Patrician Brothers to Papua New Guinea. He had been taught by the brothers at Holy Cross College, Ryde.
Second Vatican Council Doggett participated in all four sessions of the
Second Vatican Council. He would later critique its implementation though, particularly on liturgical issues. He said in 1996: "[it was] horrible, if we judge the debate on the liturgy as we have it today. Very few bishops would be proud to say they had a hand in it. Communion in the hand was never mentioned in the debate, neither was the word table (mensa) to take the place of altar, the place of sacrifice… In my opinion the debate on the liturgy has been hijacked. The Council was to reform, not to change completely." ==Retirement and death==