How is Zakat calculated on an invested sum that was not in one's possession when the hawl (Zakat year) was completed, given that a portion of it and its profits have been received? And must one await a new hawl for the profits?

You are required to pay Zakat on the principal amount and its profit once a complete lunar year (hawl) has passed over them. There is no difference between what you have received and what remains with your brother. Zakat is only due on the amount over which a complete lunar year has passed.