All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.

If the Muthan (The person who calls for prayer) said: "Allah O Akabr" this indicates that the time of prayer has started and you (Question)aren’t obliged to wait until Athan is over, rather you are permitted to initiate the prayer immediately. However, it is an act of Sunnah to wait until the Athan is over in order to repeat after it so you could gain the forgiveness that the Prophet (PBUH) promised and talked about and this won't take you more than two minutes.

Moreover, the fasting person is permitted to break his fast immediately at the beginning of the sunset Athan. And Allah The Almighty Knows Best.