They are not conservatives. They are irrational reactionaries.

Most people these days who say "conservative" mean something very different. Fifty years ago, conservatives stood for limited government, the federalist system, the balance of power between the three branches of government. Fifty years ago a Republican Congressman stated that he would have to vote to impeach Nixon, because the good of the nation came before the good of his party. That was a conservative.

Conservatives are averse to radical change. Conservatives favor free enterprise and private ownership. There's no conflict between these views and atheism. Believing or not believing in gods has no bearing on whether you're a capitalist.

Since then, the Republican party has been taken over by politicians who call themselves conservatives, but are committed to a program of radical change; the dismantling of the safety net built up since the 1930s, the rolling back of civil rights protections that started in the 1960s, the destruction of the wall of separation between church and state that was one of the principles of our nation from the beginning. They are not conservatives. And in the name of gaining and keeping political power, they allied themselves with religious fundamentalists who aren't content with believing for themselves; they want to see everyone in America and theoretically everyone in the world worshiping the Christian God. Only then, according to their beliefs, would the world be ripe for the mythical Second Coming, which also means the end of the world and the death of everyone on it in the Apocalypse.

The logical progression of welding religion into politics is the return of Manicheanism, the idea that anyone who disagrees with you is not simply arguing with you, but is against God; because, of course, you yourself are on God's side; therefore any and all opposition must be inspired by the Devil. This became extremely blatant during the Clinton administration, when TV preachers hawked videos accusing the sitting President of being a mass murderer and drug peddler, and his wife (a former Sunday school teacher, yet) of being a hateful, secret Satanic lesbian. And it's only got worse since; John Kerry (Catholic) refused communion because he was a pro-choice Democrat running for President, while pro-choice Catholic Republicans were not discriminated against; President-elect Biden, another Catholic, called "evil", "satanic", and too many epithets to be repeated here, by right-wing preachers; President Obama, a Protestant, accused of being a secret Muslim who would sell America out to terrorists.

This is the behavior of the Radical Right who simply call themselves "conservatives". There are, in truth, fewer and fewer conservatives left among the core activists of the Republican party. Those who vote in the primaries, the convention delegates, the candidates themselves, are more and more members of the radical religious right movement. They will put on a show of being concerned about the deficit and executive overreach in a few weeks, once a Democrat is again in the White House; but a show is all it will be. The conservative principles that animated the Republican party were purged in favor of a "my party, right or wrong" mentality, which most clearly showed during Trump's impeachment. The contrast with the Watergate era could not be more stark. Then, conservatives acted for the good of America. Now, the radical Republicans see only the good of their own careers and their own party's grip on power.

TL; DR: An atheist can be a conservative. What an atheist can't logically be, in my view, is a member of today's Republican party.