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At least three early adherents of apocatastasis refer to this passage. But not all their references to it are necessarily defenses of that particular doctrine. For example, Origen uses it to argue for the general love of God for his creation:
We both read and know that God loves all existing things, and loathes nothing which He has made, for He would not have created anything in hatred. [...] For He Himself is said to be the Saviour of all men, especially of them that believe; and His Christ to be the propitiation for our sins, and not for ours only, but also for the sins of the whole world. (Against Celsus, 4.28)
Clement of Alexandria's Stromata is often cited to demonstrate his belief in universalism, and he quotes this verse in it:
And to speak comprehensively, all benefit appertaining to life, in its highest reason, proceeding from the Sovereign God, the Father who is over all, is consummated by the Son, who also on this account “is the Saviour of all men,” says the apostle, “but especially of those who believe.” (6.17)
He also quotes it in Exhortation to the Heathen, which may be the earliest reference to the verse by a universalist (c. 195):
You may, if you choose, purchase salvation, though of inestimable value, with your own resources, love and living faith, which will be reckoned a suitable price. This recompense God cheerfully accepts; “for we trust in the living God, who is the Saviour of all men, especially of those who believe.” (Chapter 9)
And finally, Gregory of Nyssa cites the verse in his argument for the trinity, "On 'Not Three Gods'":
And as the Saviour of all men, specially of them that believe, is spoken of by the Apostle as one, and no one from this phrase argues either that the Son does not save them who believe, or that salvation is given to those who receive it without the intervention of the Spirit; but God who is over all, is the Saviour of all, while the Son works salvation by means of the grace of the Spirit, and yet they are not on this account called in Scripture three Saviours
There's some debate over whether these three all believe in "universalism" as it is defined today. But these three authors are certainly among the most likely candidates for that label in the early church.
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If anyone in the biblical canon should be called a Universalist, it is most certainly Paul. This is understandable, since nobody at this time had ever heard of such a thing as an "eternal hell". Paul was very mystical, mixing the Hebrew Scriptures with Greek philosophy to explain his own idea of how this was to come about.
The oldest reference to 1 Timothy 4:10 seems to come from Clement of Alexandria (150-215 AD). In the Stromata, Clement says:
"And to speak comprehensively, all benefit appertaining to life, in its highest reason, proceeding from the Sovereign God, the Father who is over all, is consummated by the Son, who also on this account is the Saviour of all men, says the apostle, but especially of those who believe. 1 Timothy 4:10"- Stromata, Book VI, Chapter 17
Later in the Stromata, Clement alludes to this passage again:
"For either the Lord does not care for all men; and this is the case either because He is unable (which is not to be thought, for it would be a proof of weakness), or because He is unwilling, which is not the attribute of a good being. And He who for our sakes assumed flesh capable of suffering, is far from being luxuriously indolent. Or He does care for all, which is befitting for Him who has become Lord of all. For He is Saviour; not [the Saviour] of some, and of others not. But in proportion to the adaptation possessed by each, He has dispensed His beneficence both to Greeks and Barbarians, even to those of them that were predestinated, and in due time called, the faithful and elect....
...And how is He Saviour and Lord, if not the Saviour and Lord of all? But He is the Saviour of those who have believed, because of their wishing to know; and the Lord of those who have not believed, till, being enabled to confess him, they obtain the peculiar and appropriate boon which comes by Him." Stromata, Book VII, Chapter 2
That is all I can find for now. I'm trying to find works from Origen, but unfortunately most of his work was destroyed [Edit: Please see Nathaniel's answer]. So this may be the only surviving Universalist use of 1 Timothy 4:10 from the early church.