Pilgrim's reply to your question is the most including yet, but I hope to add to it with mine.
Christ's divine nature was never emptied, only his glory. In Philippians 2:6, Paul says that Christ, "Emptied himself." What does this mean? That he, "Took on the nature of even a servant."
Christ took on a human nature by becoming incarnate. Simply by taking on that additional nature, he emptied himself of his godly glory. Look at his life, it was the most humble any man could ask for. To be born in a manager, to work as a carpenter, and to die on a cross.
Neither of Christ's natures was hindered in any way. Although Christ's human will was to not die on the cross, his divine will was, for he and the father are one, and it was the father's will that Christ would redeem the elect.
Christ indeed did have all the attributes of godliness, including omnipresence. "Where two or more are gathered in my name, I am between them."
