You can’t take a territory from a friendly power that way. If their capital is controlled by the enemy, you can take their territory only by capturing it from the enemy, not by simply moving into it. In your example, since the UK still would own Persia, you could move your IC there, but you wouldn’t be able to use it because the territory doesn’t belong to you.
However, if you had captured Persia from the Axis while the UK capital was in Axis hands, you would own it and could move your IC there immediately. You could also produce units there starting on the next turn after you captured the territory. Assuming that were the case, if the UK were liberated, control of Persia would return to it immediately, along with control of the IC.