If God is not constantly and simultaneously present everywhere in this natural world, how can/should we understand God's omniscience?

Omnipresence means "everywhere present," but the real question is: How is God everywhere present? God is wholly other than the created cosmos, so He is not literally in the universe. He is "in" every place in the sense that there is no place that's inaccessible to Him, no event He is unaware of. When Scripture speaks of the Spirit of God coming upon someone or being poured out, it is speaking of the dynamic presence of God—or, to put it another way, the power of God, not just a power from God, but the power of God, or God Himself acting powerfully within the world. He is always aware of all that happens as it happens, but He does not always intervene.

A good (even if imperfect) analogy is as follows: Your kids are in a large room and have access to toys, games, etc., which they can occupy their time with. The room is equipped with multiple cameras, enabling you to watch them from another room. There is also a screen and speakers in their room, so if you see misbehavior or a dispute arises, you can instantly address it. In one sense, you're already there because you're always watching, and you can immediately "be there" in the sense that at any moment, you can appear on the screen and speak directly to your kids, though you are not literally in the room.

I am in general agreement with the open theists who believe that God has full knowledge of all things present and past but does not have exhaustive definite foreknowledge—in other words, the future is largely unsettled. However, since I believe that God Himself is open (just as Scripture presents Him)—i.e., is capable of changing His mind in response to human free-will actions—I remain open to the concept that He has self-imposed limitations on His knowledge of the present and past.