What is the ruling on the prayer of an Imam who fell asleep during the first prostration of the first rak'ah, and the people alerted him, so he moved on to the second rak'ah? And what is the ruling on another person having to repeat their prayer because of this?

Scholars differed on whether sleep nullifies ablution, holding various opinions: it never nullifies, it always nullifies, its prolonged form nullifies but its brief form does not, it does not nullify if one sleeps in the posture of those praying, it nullifies only the sleep of one bowing or prostrating, it nullifies only the sleep of one prostrating, it does not nullify during prayer but does outside of it, or it does not nullify if one sleeps sitting while keeping their posterior firmly on the ground.

Accordingly, the Imam's prayer in this case was not invalidated according to the Maliki and Hanafi schools. The proper course of action was to alert him by saying "Subhanallah" (Glory be to Allah), not to interrupt the prayer. If his purification had been nullified, he should have appointed a successor. However, interrupting the prayer and resuming it sufficed for you, and the obligation was fulfilled.