JOEL - Speaking of the nature of God in the Doctrine and Covenants we read:

"He that ascended up on high, as also he descended below all things, in that he comprehended all things, that he might be in all and through all things, the light of truth; Which truth shineth. This is the light of Christ. As also he is in the sun, and the light of the sun, and the power thereof by which it was made. The light which is in all things, which giveth life to all things, which is the law by which all things are governed, even the power of God who sitteth upon his throne, who is in the bosom of eternity, who is in the midst of all things. For intelligence cleaveth unto intelligence; wisdom receiveth wisdom; truth embraceth truth; virtue loveth virtue; light cleaveth unto light; mercy hath compassion on mercy and claimeth her own; justice continueth its course and claimeth its own; judgment goeth before the face of him who sitteth upon the throne and governeth and executeth all things. He comprehendeth all things, and all things are before him, and all things are round about him; and he is above all things, and in all things, and is through all things, and is round about all things; and all things are by him, and of him, even God, forever and ever." (D&C 88)

We know our spirit looks like our body(D&C 77:2) and as such stays with our body and cannot be anwhere else, unless of course we die. God is also made of body and a spirit that looks like His body(Ether 3:15-16). Being an exalted person, He can do anything He wants but like our souls, I don't think His actual spirit personage would be anywhere but with His own body. There are no scriptures in the Bible that explicitly state that God has no body or that His spirit personage can be everywhere; they only speak of His ability to see everything everywhere and how His spiritual influence can be everywhere(Psalm 139:7-12, Jeremiah 23:23-24). God is omnipresent by His knowledge and power, and through the influence of the light of Christ and the Holy Ghost(D&C 88).

Joseph F. Smith said: "Faith in God is to believe that he is, and 'that he is the only supreme Governor and independent Being, in whom all fulness and perfection and every good gift and principle dwell independently,' and in whom the faith of all other rational beings must centre for life and salvation; and further, that he is the great Creator of all things, that he is omnipotent, omniscient, and by his works and the power of his Spirit omnipresent" (Gospel Doctrine, Joseph F. Smith 1939, 100).