What would happen if there was no influence of religion in our goverment

Lee Atwater: You start out in 1954 by saying, "Nigger, nigger, nigger." By 1968 you can't say "nigger" — that hurts you. Backfires. So you say stuff like forced busing, states' rights and all that stuff. You're getting so abstract now you're talking about cutting taxes, and all these things you're talking about are totally economic things and a byproduct of them is blacks get hurt worse than whites. And subconsciously maybe that is part of it. I'm not saying that. But I'm saying that if it is getting that abstract, and that coded, that we are doing away with the racial problem one way or the other. You follow me — because obviously sitting around saying, "We want to cut this," is much more abstract than even the busing thing, and a hell of a lot more abstract than "Nigger, nigger."

Lee Atwater (February 27, 1951 – March 29, 1991) was an American political consultant and Republican party strategist. He was an advisor of U.S. Presidents Ronald Reagan and George H. W. Bush and Chairman of the Republican National Committee.

http://en.wikiquote.org/wiki/Lee_Atwater

Policy would no longer have to be based on religious idealogy. Office holders would no longer be beholden to their religious affiliations.

I assume you meant the US

If not, nothing would change. They would just invent a religion.

I think the religious right is a big part of why we don't have universal health care, restricted funding for public education with a concurrent move to provide public funds for private schools operated by religious institutions, virtually no health services particular to women. These are the things that come immediately to mind.