Hugh Lindsay was born in Newcastle upon Tyne on 20 June 1927. He was educated at St Cuthbert's Grammar School, Newcastle upon Tyne, and Ushaw College, Durham. After priestly ordination, he worked in a variety of roles in the Diocese of Hexham and Newcastle, most notably as secretary at Bishop's House.
In 1969 Pope Paul VI appointed him Auxiliary Bishop of Hexham and Newcastle and titular Bishop of Cuncacestre. Bishop Lindsay received Episcopal Ordination in St. Mary's Cathedral, Newcastle upon Tyne, on 11 December 1969. He continued to assist the then Bishop of Hexham and Newcastle, Bishop James Cunningham, until the latter resigned the diocese in 1974.
On June 10 1977, Pope Paul VI appointed Owen Swindlehurst, another priest of the diocese, to assist Bishop Lindsay as Auxiliary Bishop and titular Bishop of Cuncacestre. The two worked together until Bishop Lindsay resigned the See on grounds of ill health. His resignation was accepted by Pope John Paul II on 11 January 1992.