This question sits on top of a mountain of assumptions based on the idea that modern life in the industrialized world is normal or universal, and that's simply not the case. So, let's unwind some of the assumptions first and then dig into the meat here.

First off, you have the relationship of the individual with time, and this varies greatly by culture and across eras. Two notable "extremes" in cultural approaches to time are so-called "polychronic" time and "monochronic" time. Monochronic time is the one most people living in the industrialized world are familiar with. A world where "time" is about schedules, daily, weekly, monthly, etc. Where every day is neatly divided into a grid of hours, minutes, and seconds. Where on that grid is laid a sequence of tasks and meetings in a precise arrangement. Where a block of time is set aside for one task (e.g. working, traveling, cooking, watching television). Where time commitments are important if not even life-critical, where a day consists of innumerable small meetings (some rigorously scheduled) with strangers for practical purposes. Where long-term deadlines and schedules are important.

This is a world built around shift based factory work (each shift works 8 hours out of 24 to ensure it is always running) and mass transit (train and subway timetables). A world built around shops and banks with set hours. A world built around urbanist interactions of strangers providing practical services to one another. The bus driver, the other bus passengers, your coworkers, the grocery store clerks, the bank teller, etc, etc, etc. Monochronic scheduling is designed to facilitate these casual, practical interactions. The bus, the grocery store, and the bank all have set schedules so they are predictable and able to be integrated into people's personal schedules based on factory work.

Contrast this with the polychronic time experience. A world where relationships and people are important, where most interactions are with people who aren't strangers, who are family, friends, members of a close-knit local community, people you interact with in innumerable ways over years and years, and with whom you have complicated interpersonal relationships that extend much beyond mere temporary, practical interactions. Where tasks aren't divided neatly into a gridded schedule, where people typically are doing multiple things at the same time (simple example: doing the laundry, baking bread, and preparing produce for dinner). People who have this sort of relationship with time tend to be derided as unserious by monochronic folks, and this behavior has been given many derogatory names from "island time" to all manner of racist variants I won't mention.

The frustration of monochronic time experiencers with polychronic individuals is understandable, to them it looks like laziness, unseriousness, and backwardsness. It seems perfectly obvious, and indisputable that time must be a strictly scheduled grid, that temporary, practical, rigidly scheduled relationships are the foundation of social order, etc. But looked at another way, monochronic time involves a lot of very dubious tradeoffs. It presupposes a life that is constantly rushed and harried; built around innumerable meaningless, shallow, and mercenary relationships; where success is measured in the form of materialistic acquisition of status symbols, power, and hedonistic luxury goods. In contrast, the polychronic view of life as a web of close, deep, and long-lasting familial relationships certainly seems a lot more enlightened.

A polychronic approach to time is the more natural, innate one in humans, and for most of history was the dominant approach to time. Anyone with a pet or an infant knows that being able to tell time isn't necessary to have a daily routine. The monochronic approach to time built itself slowly over millenia. Some early hints at it came in the form of "watches" for guard duty or on sailing vessels. With the advent of industrialization and especially steam and then electrical power you had (as alluded to above) factories which operated in fixed shifts. There it was important for everyone to work together for a set number of hours. One of the most important factors in the advancement of monochronic time was the railroads. Even when it was common for many people to have access to "the time" via public clocks, church bells, personal watches, etc. these were still generally not universally synchronized. In, say, 1850, for example, "the time" in New York City would be different from the time in Boston, someone making a trip between cities would need to adjust their personal watch (if they had one) to the local time. It wasn't until the late 19th century that the concept of a universal "railroad time" and of fixed "time zones" became widely adopted, cementing monochronicity into the structure of daily life.

As for peasants, the answers here are pretty straightforward. How did they know when to get up and go to bed? They didn't have alarm clocks, but they did generally live more communally than we do in the industrialized world today. Most people would wake up to the sun, and if not that then the commotion of people getting started with their day would wake them up (or, someone poking them and telling them to get out of bed). Every day has its own natural rhythm. When the sun sets things start winding down, you have a routine of various things that you do at night before bed, then you go to sleep when you feel tired or when everyone else turns in. During the day you might have things to do for your job as a peasant, and you either do these with others or you do them alone, but there's generally a routine or a set of routines that organize people's day and as a consequence the work gets done without the need to schedule everything down to the minute. Maybe you get up and milk the cows, then afterward you feed the chickens, and so on. Some days might involve a lot of people working together, which would be more organized, but the general shape of such things is still fairly organic. People aren't commuting into their jobs so getting everyone together to do something together was just a matter of rounding people up and starting when that was done. Depending on region and era there might be different schedules throughout the year. Different days of the week when people were expected to work or have time for themselves (if any). Different periods of the year when work would be more or less intense or involve different sets of tasks.

/u/yodatsracist has previously answered How did people wake up before alarm clocks?

/u/frogbrooks has previously written another answer about waking up early for prayers in times past. .