Don't most jurisdictions have an alternative to an oath for those whose religious convictions keep them from taking an oath? Quakers and Mennonites, at least, I think, do not swear oaths.
My jurisdiction says "on penalty of perjury" instead of "so help me God" as part of the general swearing-in process, so I haven't seen anyone make a religious objection, but I also haven't come across a non-oath-taker yet, so I'm not sure what happens here.