Most people just rose at dawn, or when the birds started their noise, like a rooster. Those few who really had to wake up earlier were usually woken up by 'specialists', like the Knocker up
Letting someone else wake you is just shifting the problem. How would those people wake up?
As long as a human keeps a fairly regular circadian rhythm with sleeping at night then there are several factors or biological mechanisms built-in, our own time keeper, synced to the Zeitgeber:
- after your amount of necessary sleep, you wake up
- when daylight comes, you wake up
- when you plan on having a certain time to wake up, you tend to wake up shortly before that time has come
That is of course not a guarantee for waking up if you have to.
That's a problem common enough that even the ancients Greeks turned to artificial inventions. We can be sure that those who built the Antikythera mechanism had something up their sleeves.
Do not look further than Plato as the inventor of one of the first alarm clocks:
Video on YouTube from a Greek museum
The alarm clock of Plato
(the first awakening device in human history)
The alarm clock of Plato:
The upper ceramic vessel supplies the next vessel through an (appropriately calculated for every case) outflow funnel.
When the second vessel becomes full at the programmed moment (for example after 7 hours) through the internally located axial pipette, it evacuates fast towards the next closed vessel and forces the contained air to come out whistling through a tube at its top.
After its function, the third vessel empties slowly (through a small hole located at its bottom) towards the lower storage vessel in order to be reused.
Archimedes and Aristotle also had their go at clocks.
So, they all did the same we do today from a certain point in time forward: either wake up naturally, or use an early form of alarm clock, if they could afford one.
And yes, before the industrial age and cheap artificial lighting, people had a much healthier sleep cycle, even if it was fragmented, for whatever reason.
A. Roger Ekirch: "Sleep We Have Lost: Pre-Industrial Slumber in the British Isles". American Historical Review, 106:2, 2001, pp. 343-386.