Theology, of course, means the study of God. Everyone has a theology: the atheist fervently believes in the absence of God as much as the man fervently praying. So a bunch of Non Christian (TM) theological songs.
Let’s start with a Sufi (Richard Thompson) writing something covered by the great Norma Waterson.
Nick Cave, like Leonard Cohen, is known for writing both religious and controversial songs. Most people would say “In my arms” is theological since it begins “I do not believe in an interventionist God / but I know darling, that you do“. But this is as philosophical.
I am going to end this, however, with some Serbian Chant. Divna Ljubojevi? is Orthodox, Serbian, and demonstrates what minimalism should be like. She is also, generally, singing the liturgy.