Does divorce take effect upon the wife's arrival in Egypt, making it incumbent upon the husband to divorce her, after he swore to that effect saying: "By God, when you go down to Egypt, I will divorce you," bearing in mind that he does not desire the divorce and was trying to calm her down?

It appears that your statement to your wife, "By Allah, if you go down to Egypt, I will divorce you," is an oath of divorce conditioned on her going down to Egypt. If you do not execute the divorce, you have broken your oath, and an expiation for breaking an oath becomes incumbent upon you. Breaking the oath here is preferable if holding onto your wife is better, as the Prophet, peace and blessings be upon him, said: "Whoever swears an oath and then sees something better than it, let him do that which is better and expiate for his oath."