The LDS hold microfilms of the following records, which can be ordered and viewed at one of their Family History Centres for a fee:
Church of England records
Bishop's transcripts for Gibraltar Anglican Chaplaincy, 1807-1812. Rites performed by Anglican clergy for British military stationed on Gibralter. (Outside the time period you're interested in though I know)
Roman Catholic
Parish registers, 1704-1991. These cover births, marriages and deaths for the following churches:
Catholic Church. Cathedral of St. Mary the Crowned (Gibraltar)
Catholic Church. Sacred Heart of Jesus (Gibraltar)
Catholic Church. Our Lady of Sorrows (Gibraltar)
Catholic Church. St. Theresa (Gibraltar)