I've heard a lot of atheists say that atheism is not a worldview because atheism doesn't make any claims about the world. Atheism is only a lack of belief in God, which is not a positive claim.
While I'm perfectly happy to accept the definition of atheism as "lack of belief in God," I still think it's meaningful to speak of an "atheistic worldview." It seems to me that any worldview that lacks a God is an atheistic worldview. I would agree with atheists that there's no such thing as the atheistic worldview since an atheist might be a naturalist, an idealist, a pan-psychist, and even a substance dualist. There's not a whole lot you can tell about a person's worldview just because it doesn't include a god. Still, if a person has a worldview, and that worldview does not contain a god, then I would call that an atheistic worldview.
Since many atheist say that they aren't making positive claims about the world by virtue of being atheists, there can't be an atheistic worldview in the sense of being a worldview predicated on the non-existence of God. I'm not sure I agree with that. Here's an analogy. If you remove sunlight, things will get dark and cold. So a world with sunlight is different than a world without sunlight. Lack of sunlight doesn't positively cause anything since there's nothing to do the causing. But there is still a sense in which lack of sunlight makes a difference to how things are. In the same way, it seems to me there are consequences to there not being a God, and those consequences can inform a person's worldview.
I don't think, for example, that there could be objective moral obligations if there were no God. A lot of atheists agree with me about that. They don't believe there are any objective moral obligations. Whether there are objective moral obligations are not is certainly something relevant to a worldview. So it seems to me that any consequence of there not being a God would amount to something an atheist ought to affirm even if they don't affirm anything about the existence or non-existence of God. If God is lacking in their worldview, then their worldview doesn't contain a God, and if their worldview doesn't contain a God, then they should affirm whatever the consequences of how reality would be if there were no God.
There's still room for disagreement about what those consequences would be, of course. Some atheists don't think there are any consequences regarding morality either because they don't think objective morality requires God, or they don't think God would be sufficient for objective morality even if there were a God. Nevertheless, it seems like almost anybody could agree that there are some consequences to there not being a God. With that being the case, maybe we could say that an atheistic worldview is something a little more robust than simply a worldview that lacks a God. We could possibly make some positive claims about that worldview.
One way an atheist might work out such a worldview is to think about what difference it would make to reality whether there were a God or not. I suppose if the atheist couldn't think of anything, then he wouldn't be able to make any positive claims about reality just by virtue there not being a God in their worldview.
Christian apologists, on the other hand, can attempt to argue that there are entailments to there not being a God in a similar way that they might argue that there are entailments to there not being a sun. If an atheist says they lack a God in their worldview, but then they either affirm or deny the logical consequences of there being a world without God, then Christians can accuse them of being inconsistent. Maybe atheists don't realize they are being inconsistent, or maybe they just disagree with the Christian about what the logical consequences are to a world without God. But that would be where the debate would lie.
That seems to be the strategy of the presuppositionalists. They believe things like logic, math, minds, morality, laws of nature, etc. would not exist if God didn't exist, and since atheists affirm most of those things while not believing in God, they are being inconsistent. The strategy then becomes trying to get atheists to realize it so they'll either affirm the existence of God or face being irrational. Atheists, on the other hand, can respond by either denying that God is necessary for these things, or by biting the bullet and admitting that logic, morality, etc. aren't real.
I really think this dispute about whether or not atheism is a worldview or whether there's such a thing as an atheistic worldview boils down to semantics. Usually when I'm arguing with somebody, and I realize we're just arguing over the meaning of a word or phrase, I try to avoid using the word altogether. I try to get the other person to talk about the substance rather than the word.
Not long ago, I had a conversation with an atheist, and I was trying to find out what he believed about God. I was just trying to find out whether he had any opinion at all on whether or not God existed. Instead of answering my question, he just kept reminding me of what the definition of atheism was. I finally told him I didn't care about the definition of atheism. I just wanted to know what he thought. Was he 50/50 on the existence of God, or did he think there was no God? Telling me he's an atheist and that atheism is a "lack of belief in God" doesn't answer that question. It was the substance of what he thought that I was interested in, not the meaning of the label he placed on himself. Arguments over the meaning of words are just distractions.