(Disclaimer: I'm Catholic, speaking for me, but I think that this agrees with the main official/majoritarian catholic belief)
Even if we could read into the mind of person, to know his most sincere and precise beliefs, we could never deduce from that he will be saved or not. In principle, it's perfectly possible - as long as the atheism of the atheist consists in non-believing in a "wrong" God (which, I'd say, is quite frequent).
But we just don't know. All we (speaking for christians) know in this respect, is "practical knowledge", not "theoric knowledge": it is directed to our acting (in the broad sense of the word, see below) and not to our speculating. I know that my belief in Christ is necessary for my salvation, and that's all.
I think that your dychotomy is flawed: "if the deeds you perform in life might also get you into heaven, even if you don't believe in the Christian God"
You seem to assume that faith is just one intelectual act (like an opinion), and your other human acts can be put apart. The "belief" of what Jesus speaks is surely something more deep that an intelectual assertion (which is out of reach of lot of people who haven't heard a word of Jesus, or that have heard inept words; not to speak of dements, babies, etc), it's rather more akin to some cosmic attitude, a sense of reverence at the Goodness/Truth/Beauty, and the sense of sacrifice, the active will of "working" for them. This "belief" is "doing the will of the Father", and it's not some bunch of intelectual assumptions that a person has in his mind, while his acts can go their own ways: it's inseparable from acts (in the broad sense: external and internal; including perhaps religious acts; and including above all loving your neighbours -charity), it's almost immanent to them. Of course, believing in Christ, and in general the religous faith (more precisely, in catholic terms, the three theological virtues: faith, hope and love), is both the higher "act" and also can well be a necessary act... depending on the circumstances of each person.