Five key things need to be held together here.
1) The disciples asked Jesus two particular questions that touch on our subject: “Lord, why do you intend to show yourself to us and not to the world?” (John 14v22) and “Lord, are only a few people going to be saved?”(Luke 13v23). In both instances Jesus’ answer was hardly encouraging. He stresses the need of his hearers to ensure they believe. This issue cannot therefore be used to sidestep faith in Christ. The fact is, we have heard and are responsible for accepting or rejecting him. Indeed, as John makes clear, "God's wrath remains" on those who hear of Christ yet reject him (John 3v36).
2) God is the author of salvation. We can therefore be sure that he can ensure the gospel is heard by all those who will be saved. Consider the vision given Paul or his sending Peter to Cornelius as just two examples.
3) One can only do justice to the New Testament data by concluding that we can have no assurance that anyone is saved without conscious faith in Christ. Evangelism must therefore be our priority. Two particular passages are pertinent here:
"For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life. For God did not send his Son into the world to condemn the world, but to save the world through him. Whoever believes in him is not condemned, but whoever does not believe stands condemned already because he has not believed in the name of God's one and only Son.”
John 3v16-18
Everyone who calls on the name of the Lord will be saved." How, then, can they call on the one they have not believed in? And how can they believe in the one of whom they have not heard? And how can they hear without someone preaching to them? And how can they preach unless they are sent? As it is written, "How beautiful are the feet of those who bring good news!"
Romans 10v13-15
We just cannot read these verses and still buy into the contemporary assumption that all who are sincere in whatever belief will be saved.
4) Having said that, there are sufficient allusions within the wider scriptures to say that God’s saving grace may (and we can say no more than 'may') deem those who believe in and call on the God of Israel for mercy as effectively believing in and calling on Christ. Here we would say that the nature of God as trinity means that the fact they do not know this God in the man Jesus does not mean that they are not responding to God the Son when they respond to God.
We might consider characters like Melchizedek, Jethro, Rahab and the Queen of Sheba. They do not seek salvation through false religion, and on the basis of a very limited knowledge their acknowledgement of God is commended. Rahab in particular is mentioned in the line-up of Hebrews 11 as someone of faith.
Perhaps most significant here is Nineveh under the preaching of Jonah. The text is intended to express the wideness of God’s mercy, so we should be cautious about dismissing it, and doubly so when we consider Jesus used the Ninevites as a model of repentance (Luke 11v32). What we see in them is (a) an acknowledgement that the God of Israel is the God of heaven and earth, (b) a calling upon him for mercy, (c) a change of behaviour that follows. Now we can hardly doubt that this patterns the gospel of repentance and forgiveness of sins in Christ’s name (Luke 24v47).
All this would make John 3 and Romans 10 (above) either encompass such belief as belief in the Son, or simply describe the general lostness of the world where people need concious faith in Christ and cannot attain it without hearing of him. If the latter, this would require us not to push the texts to apply to everyone without exception who hasn't heard.
Whatever the case, we must be clear that any such salvation would be despite the person’s inherited religion (or beliefs). It would be because God has enabled them to pick up enough of the himself through what is widely known of him through the influence of Judaism and Christianity that they acknowledge him as their Creator and Judge, and so repent with deep sincerity and cry to him alone for mercy. We might imagine then the truth of the gospel radiating like sound waves from his church. Those close hear everything with clarity, those far away only hear an uncertain whisper, but still may hear enough to respond.
We should recognise that doing so would mean that they reject the systems of salvation within all non-Christian religions (including Judaism and Islam) because, in contrast to the gospel, they stress reliance on one's own religious or moral works for God's acceptance. Moreover, we should also recognise that the test of whether such a person has responded to the true God would be that when faced with Jesus and the gospel, they would readily accept them. As John says, "whoever lives by the truth comes into the light, so that it may be seen plainly that what he has done has been done through God" (John 3v21).
We might understand Peter's interaction with Cornelius in Acts 10-11 in this way. This would make the Spirit's testimony that Peter will bring "a message through which you and all your household will be saved" as suggesting not that Cornelius wasn't already saved through his faith, but that either (a) his salvation would be completed with the gift of the Spirit through the message, or (b) he would now hear the message of how his salvation was made possible.
It must be stressed, however, that we just do not see people throughout the world responding with such heartfelt repentance from sin and reliance on God alone for mercy. We cannot therefore be complacent and assume all will be well with them. Instead we must step up to our commission to bring the clarity of the gospel to them so they would definitely respond as God calls them to.
5) A final question is whether people can be saved by God enabling them to glean sufficient knowledge from the created order. On this, at best we must be agnostic. Romans 1 tells us that although we suppress the truth about God, all humanity still instinctively know there is a Creator (v20-21) and that we deserve punishment at his hand (v32). And elsewhere Paul asserts that the blessings of creation testify that God is a God of grace (Acts 14v17). Because we are truth supressors, we cannot understand this of ourselves. But we cannot say it is impossible for God to give us this understanding by his Spirit. However we have no inferences of faith in scripture apart from picking something up from God's special revelation through Israel or Christ and the apostles (or in the case of Melchizadek, by tradition perhaps from Noah). Therefore we have to say that there is no suggestion that God does save people through the witness of the created order alone.
Conclusion:
We must summarise to ensure that I am not read to be saying something I'm not:
1) We can have no assurance that anyone is saved without concious faith in Christ.
2) Some may be saved through what is gleaned of God by the impact of Judaism and Christianity on the world, but only if they reject all works based ideas of salvation to acknowledge the God of these religions as the true Creator God, call on him for mercy and turn from sin.
3) There is no suggestion that people will be saved simply from what they witness of God in the created order.
4) The emphasis of the scriptures is on the need to hear of Jesus, and so we would be foolish not to ensure we respond ourselves and negligent if we fail to urge others to also.