The recordings were originally made for a movie. A full length soundtrack for the movie that never existed is available here.

Tramp with Orchestra IV (full strings) [MP3]
Tramp and Tom Waits with Full Orchestra [MP3]

When I copied the loop onto the continuous reel in Leicester, I left the door of the recording studio open (it opened onto one of the large painting studios) while I went downstairs to get a cup of coffee. When I came back I found the normally lively room unnaturally subdued. People were moving about much more slowly than usual, and a few were sitting alone, quietly weeping. I was puzzled until I realized that the tape was still playing and that they had been overcome by the old man’s unaccompanied singing.

From Wikipedia:

Jesus’ Blood Never Failed Me Yet is a piece of music composed by Gavin Bryars in 1971. It is based around a recorded loop of an un-named tramp improvising a hymn; eventually rich harmonies are played by a live ensemble of strings and brass, always increasing in density. The recording was originally made for a 1960s documentary by Alan Power which chronicled street life in London. Later when listening to the recordings, Bryars noticed that the clip was very in tune, and that it looped well into 13 bars. The tramp died before being able to hear the completed piece.

Links:
Wikipedia Article
Gavin Bryars
More about the recording