As I have said in the comment as well, "The only wise God" does not necessarily imply other deities who are not wise, but this wording could be correct also in case no other deities existed at all, or, even if they existed, still with no reference to them. If something exists as unique in its kind, say, "the only genius painter of Mona Lisa", with saying this sentence, I do not imply at all that there are other, non-genius painters of Mona Lisa, but just praise Leonardo da Vinci for what he is, without any competitive/comparative reference to other painters. However, contextually, if it is commonly known that, for example, Mona Lisa was also painted by less gifted artists, the same phrase could indicate to such a competition/comparison. Greek of the sought passage is open to both interpretations.
However, at a deeper theological level one can say, that "wise" is said here in an absolute, universal and not relative, particular sense, as an attribute of somebody, who cannot be not-wise, who is always perfectly wise, and whose wisdom was a foundation of all creation (Prov. 3:19; Psalm 104:24). Now, such a never shaken and always perfect wisdom is a property only of God, while no human or angel can have it.
In this sense, it is also important to notice that the phrase "the only wise God" is preceded in the same sentence by the "αἰωνίου θεοῦ" - "the eternal God", that is to say, in order to intimate that the wisdom of God is as eternal as God himself, or to put it otherwise, God is eternally wise. Humans can by no means have such a wisdom properly, but only by participation; in this sense, Jesus is also possessor of the same unshakable wisdom, which believers possess not properly but by participation (1 Cor. 2:16), while Jesus himself surely possesses it properly, just as God does. And just as God cannot deny himself and be otherwise than God, so also Jesus, in the identical way, cannot deny Himself and is always unshaken, when the greatest of men (and even angels) can be shaken and fall (2 Tim. 2:13), for that very reason He is the Principle of judgment and forgiveness.
The context of the Romans 16:25-27 also indicates that the mystery hidden from the beginning of the world, i.e. God's mystery, which semantically can be closely related to if not identified with His wisdom, is revealed through Jesus Christ, and then in the 16:27 also, διὰ Χριστοῦ can be understood as "God who through Jesus Christ has shown Himself to be alone wise (in the absolute sense)" (see: Mayer's NT commentary http://biblehub.com/commentaries/romans/16-27.htm) - thus, like the mystery is revealed through Jesus Christ, so also His being the only wise God is also revealed through Jesus Christ; and Paul is emphatic here that God is absolutely powerless to reveal His being wise but through Jesus, who as necessary and co-eternal principle of this revelation is elsewhere also called by Paul the very Power and Wisdom of God (1 Cor. 1:24).