At 1 Ti 1:1 we have a perfect example of what Daniel Wallace calls the “anaphoric article,” where “the first mention of a substantive is a anarthrous.” This is common in the salutations of NT books when someone “ is introduced.” 
“God” in “God our Savior” and “God the Father” in the introduction of 1 Timothy both lack the definite article and are therefore anarthrous.
1 Ti 1:1 Paul, an apostle of Christ Jesus according to the commandment of God our Saviour, and Christ Jesus our hope; 2 unto Timothy, my true child in faith: Grace, mercy, peace, from God the Father and Christ Jesus our Lord (ASV)
Right before 1 Ti 2:5 in verse 3 the same expression “God our Savior” is repeated. This time it has the article and therefore is one of the “subsequent mentions” that identify “God” as the Father in the book.
The linguistic reason for this identification is that the Greek anaphoric article is usually equivalent to the pronoun. 
1 Ti 2:3 This is good and acceptable in the sight of God our Saviour; (ASV)
Therefore at 1 Ti 2:5 “God” is identified as the Father from verse 3, and distinguished from Jesus in the work of salvation.
1 Ti 2:5 For there is** one God**, one mediator also between God and men, himself man, Christ Jesus. (ASV)
This does identify the “one God” as the person of Father.
For more information on the anaphoric article as applies to other verses with θεός see my paper “The Greek Anaphoric Article Applied to the Exegesis of 2 Peter 1:1 and Related Texts – A Fresh Grammatical and Contextual Analysis”
 "The anaphoric article is the article denoting previous reference. (It derives its name from the Greek verb άμαφερειν, “to bring back, to bring up.”) The first mention of the substantive is usually anarthrous because it is merely being introduced. But subsequent mentions of it use the article, for the article is now pointing back to the substantive previously mentioned. It is the most common use of the article and the easiest usage to identify." .... “Practically speaking, labeling an article as anaphoric requires that it have been introduced at most in the same book, preferably in a context not too far removed. (Wallace, Greek Grammar Beyond the Basics An Exegetical Syntax of the New Testament, p. 217)
 “it becomes evident that there is no ground whatever for making a distinction between the nature of the Article o
and the Pronoun o and that the “near relation” is in truth no other than perfect identity.” (Middleton, T. F. (1833). The Doctrine of the Greek Article, p. 13.)