Does 1 Tim 4:10
10 For therefore we both labour and suffer reproach, because we trust in the living God, who is the Saviour of all men, specially of those that believe.
Does this verse teach that God in Christ is the Saviour of all men without exception? By no means, that is a contradiction of scripture, lost men don't have God as a Saviour, duh!
Instead the verse teaches limited atonement just like the Truth is in Christ Jesus. The word especially here is a adverb and it actually is a qualifier of who God in Christ is a Saviour unto. The word means:
most of all, especially.
málista (from mala, "very much") – very much the case; particularly so; especially (mostly) so.
See that, its teaching particular salvation, Hes a Saviour particularly to them believing. The word especially also means, exclusively. So God is Saviour exclusively and particularly to believers, whether jew or gentile.