What is the ruling on someone other than the Imam leading congregational prayers in the mosque when the Imam is absent or late? And what are the conditions for its acceptance?

If the Imam is late or absent, and one of those present leads the people in prayer, their prayer is valid. The Companions prayed behind Abd al-Rahman ibn Awf when the Prophet, peace and blessings be upon him, was late, and the Prophet approved of them saying: "You have done well" or "You have acted correctly." There is no objection to waiting a short while for the Imam if time permits; otherwise, waiting is not permissible.