How did Imams wake up in time to make the call for morning prayer before alarms existed?

In the medieval Islamic world, there were two primary offices associated with regulating the times for prayer: the muwaqqit, in charge of calculating the times for prayer; and the muezzin, in charge of announcing it was time for prayer (and other duties).

As you can imagine, determining and calling the appropriate prayer times was of deep concern for medieval Muslims, so scholars produced enormous treatises on various methods of calculating the hours. Astrolabes and sundials were among the more precise methods. Scholars produced books of tables calculating prayer times throughout the year or at different latitudes. Texts waver between denouncing the "uneducated" muezzins who rely on the angle of shadows during the day, and teaching muezzins how to use the angles of midday shadows to announce prayers.

Medieval Christian monks, who also had to keep time throughout the night to pray the Divine Office at appropriate times, also had someone standing watch throughout the night. I mention them here because Latin Europe is where I'm familiar with the procedure for cloudly nights. When the sky was obscured, medieval monks and nuns kept time by chanting psalms. My educated guess is that in similar situations, medieval muezzins would have chanted suras or prayers.

So the observant muezzin had a good deal of information in front of his eyes to know when to announce prayer. So how to ensure an observant muezzin at all times?

Simple: have more than one. When we look at salary records for mosques and khanaqahs (Sufi "lodges" or group houses/schools), which are sometimes included in foundation/endowment documents, we find one person endowed/salaried to teach Quran, one person endowed to teach one school of Islamic law, one person endowed to teach another school. Muezzins? Always multiple. The 1386 waqqifiya or foundation document of a Sufi madrasa in Cairo, for example, provisions six to eight muezzins. They would simply rotate timekeeping duty around the clock.