Truly religious street-performers may or may not be asking for money, and if so, it may not be unambiguously clear that any contributions are payment for their music rather than for their church or whatever, but I can't imagine that they might not "count" as buskers.

It takes the same amount of nerve to put yourself out there on the street whether your repertoire is secular or religious, right?

One of my very favorite groups of fellow street-performers in New Orleans in the early '70s was a trio of elderly African-American gospel singers. They had appeared a year or two earlier in the movie Easy Rider ~ they are seen on-screen only very briefly, but their music continues in the background for quite a while afterwards, all through the acid-trip cemetery scene. I have often wondered if they had been appropriately compensated...