1 John 4.14 says: And we have seen and do testify that the Father has sent his Son as the Savior of the world.
Describing Christ as the "Savior of the world" is rare in the bible. It appears only here and John 4.42.
Likewise at 1 Tim 4.10 we find: For to this end we toil and struggle, because we have our hope set on the living God, who is the Savior of all people, especially of those who believe.
These are very powerful statements about the universal scope of God's salvation in Christ. The Son of God is the savior of the world (σωτῆρα τοῦ κόσμου) and the savior of all human beings (σωτὴρ πάντων ἀνθρώπων). These are statements which emphasize both the universality of Christ's salvation as well as its actuality.
It wouldn't be plausible to read these statements in conditionalized ways, as if what John meant is this: the Son is the savior of the world, in that if the world is going to be saved at all, it would be through him. Likewise Paul's words to Timothy cannot mean something like this: God is the savior of all people, in the sense that if people are going to be saved at all, it has to be through him. These readings are not plausible for a number of reasons.
To begin, John is describing the reason for which the Father sent the Son into the world. He sent him in to be the world's Savior. But if we accept the proposed conditionalized reading above, then he wouldn't need to have sent him at all. Christ could have remained in heaven, never having come to earth, and he would still be the "savior of the world" in the sense proposed, because it would still be true that the world would only be saved through him if at all.
A similar line of reasoning can be brought forth against the proposed conditionalized reading of 1 Tim 4.10. If God does not actually save some persons, then in what sense can he be called their savior? Suppose you and a group of your friends are taken hostage by ISIS, and American soldiers come. You exclaim, "Finally, my saviors have arrived!" Then the soldiers proceed to free some of the hostages but not others, leaving you behind. They leave, and then the ISIS operatives return from their lunch break and proceed to torture you out of frustration. Would it make any sense to call those American soldiers your saviors? To my mind it would seem not; if they were your saviors, they would have saved you. So also, if God is the savior of all human persons, then it seems he does save all human persons.
There is a further complaint to make about these conditionalized readings. In 1 Tim 4.10, Paul uses the verb εἰμί (to be) in the indicative mood: God is (ἐστιν) the savior of all human beings. Verbs in the indicative mood refer to actual realities: John is writing; Paul is speaking; Jesus is preaching; etc. This conditionalized reading of the passage negates the indicative force of the verb, however, by adding a hypothetical where the verb doesn't communicate one: God could be the savior of all persons. This reading, to my mind, is grammatically problematic.
It seems to me clear enough that in the scriptures, a savior is a person who actually saves or will save. Consider these examples of the phrase as it is used in various passages:
The Lord is my rock, my fortress, and my deliverer, my God, my rock, in whom I take refuge, my shield and the horn of my salvation, my stronghold and my refuge, my savior; you save me from violence. I call upon the Lord, who is worthy to be praised, and I am saved from my enemies (2 Sam 22.2-4).
Therefore the Lord gave Israel a savior, so that they escaped from the hand of the Arameans; and the people of Israel lived in their homes as formerly (2 Kings 13.5).
Therefore you gave them into the hands of their enemies, who made them suffer. Then in the time of their suffering they cried out to you and you heard them from heaven, and according to your great mercies you gave them saviors who saved them from the hands of their enemies (Neh 9.27).
They forgot God, their Savior, who had done great things in Egypt (Ps 106.21).
It will be a sign and a witness to the Lord of hosts in the land of Egypt; when they cry to the Lord because of oppressors, he will send them a savior, and will defend and deliver them (Is 19.20).
For I am the Lord your God, the Holy One of Israel, your Savior. I give Egypt as your ransom, Ethiopia and Seba in exchange for you (Is 43.3).
The title of savior comes hand in hand with saving activity. Consequently, if God is the savior of all human persons, and if Christ is the savior of the world, then all human persons and the world will be saved.