It's not entirely true that atheists and churches are wholly incompatible with one another; Unitarian Universalist congregations can be a mixture of atheists, agnostics, and theists. And as an offshoot of liberal Christian denominations, the UUs retain music-inclusive roots and traditions.
An artist of interest in this line would be Cat Faber, whose work is most often grouped under "filk" but who also very much qualifies (IMO, anyway) as a modern folk performer and whose lyrics often display strong humanist elements. (See also the Bandcamp page for Echo's Children, a now-inactive duo of which Cat was half.) I should mention that the Bandcamp pages are considerably more up-to-date than those on the artists' outlying Web sites.
A couple of specific examples:
• Atheist's Anthem
• Song of the Makers
Scattered among the albums and single track releases are a number of non-religious songs I'd consider wedding-appropriate, including one or two specifically titled as such.