Is it obligatory to interrupt the prayer—whether it be a Sunnah or a Fardh—upon hearing my brother's adhan after having started it, and am I obligated to wait for his adhan no matter how long it takes, especially if I wake up early and he calls the adhan late, after the time for prayer has already begun?

It is recommended for someone praying alone to pronounce the Adhan (call to prayer) and Iqama (second call to prayer) due to the merit of the Adhan, even if they enter a mosque where the prayer has already been performed. However, a woman should pronounce the Adhan for herself in a low voice that cannot be heard by non-mahram men. If you are praying a fard (obligatory) prayer and are certain that the time for prayer has begun, then you hear your brother's Adhan, complete your prayer and do not interrupt it, nor should you repeat the Adhan during it. If you are praying a nawafil (supererogatory) prayer and then hear the Adhan, do not interrupt your prayer. It is permissible for a woman to pronounce the Adhan immediately upon the entry of the prayer time without waiting for her brother's Adhan. If it is possible to gather for congregational prayer, it is better for both men and women.