That is Roger Lloyd Pack's real singing voice.
"Songs of Praise" is a real BBC Television religious program that presents Christian hymns which first aired in October 1961, and the series is one of the longest-running of its kind on television anywhere in the world. Usually it is shown on various times between late afternoon and early evening every sunday. During the period this episode of "The Vicar of Dibley" was made, the format version of "Songs of Praise" included congregations from churches and cathedrals singing hymns whilst the presenter explored that week's theme, all from the same location.
Owen mentions Aled Jones when auditioning , Aled Jones now hosts Songs of Praise.
Geraldine tells Alice to 'Read on, Macduff', a spoof of 'Lead on, Macduff' from Rider Haggard's 'King Solomon's Mines' (1885), which is itself a pun of 'Lay on, Macduff' from Shakespeare's 'Macbeth'.