Whenreading fantasy novels, I have as a rule been left with the impression that exact timekeeping is either avoided as a topic, or referred to only vaguely and obliquely. In many cases, the author would freely use "hours" as a metonymy for a part of the day spent doing something, however never actually imply that these hours can be counted and consist of minutes and seconds, or let his elven queen invite the orc delegates to convene in her throne room at precisely 10.30 in the morning for trade negotiations.
As several have noted already, sexagesimal numbers are exceedingly convenient for quick calculation, because 60 is divisible by 2, 3, 4, 5, and 6. This naturally lends itself to divisions of 12 & 5, thus the division of our clock face into 12 blocks of 5 minutes, our 12 months of the year, and so forth.
I suggest that you keep to something like this. The obvious alternative is a lunar calendar, as traditionally used in China and its broad area of influence (among other places), but you have to understand it clearly enough that you can use it fluently and fluidly in passing conversations among characters.
You now have a complete system of decans. These label every ten-degree arc of a complete circle: lion-head, daisy-stem, etc. Under normal circumstances, however, everything will be labeled by the main image (hour of the lion, month of the daisy, etc.).
First you need to understand that the concept of time that we have today is only about 200 years old. We didn't have time that was shared with more then a handful of people until the popularity of the train made it a requirement. Until that time, a community would have one "master clock" that everyone else would set their "little" clocks by.
Even then, modern time keeping didn't exist till maybe 50 years ago, when it became more important for two parties to do a thing at the same time. Computers and other time pieces that share information ( the ability to sync a clock) is very very new, even in our world.
Back when a community ran on a shared clock (usually at the town hall or other important civic building), people would set a schedule by it. However that schedule was only important to the people in range of that clock. Everyone else just used a generic time like "Next Tuesday Morning" instead of 9am Tuesday.
Now back in your world, what you need is a reason to have time exist in more then one location. The real trick is why do you need that, and how are people going to track it. How are they going to sync it? What needs to be done at such a precise time that it's worth that effort? Remember that until very recently, everyone not sitting next to a community clock still just used sun up, noon, morning, etc. And two people sitting next to different clocks could be hours off.
I think we all agree that the purpose of clocks is to synchronize the activity of multiple people over an area. We like to say things like "At 12:30 I will go to the store." However, is is also possible to synchronize actions based on events. "I will go to the store after the Blacksmith I am apprenticing finishes eating his lunch." One can synchronize that way.
Consider the courts in fantasy capitols. Time can be managed by the flurry of movement of paiges letting people know how the audience with the queen is going. In fact, in many cases, this is a more effective approach to timekeeping. How many times have you had a 30 minute meeting that becomes a 4 hour discussion? It happens often enough that flexibility in time is useful.
This works well until you need people who aren't interacting to all convene at the same time. Then you need something more synchronous. Often the town bells may be used to provide such a synchronous moment for all people. In Islamic countries, they could syncrhonize to the Muezzin calls, 5 times a day.
In the end, clocks are needed when you must synchronize many arbitrary events at many times of day, over long distances. That calls for an absolute sense of time, and if you need that, you need a clock. However, if your fantasy culture doesn't need quite that much synchronicity, time may be defined by particularly important events to the people of the area. Many school children learn to tell time of day by how many recess bells they hear, long before they learn to read a clock. Such a culturally specific form of synchronization could fit the bill.
In the other extreme, farm life is not all that dependent on exact timing. You may find that sunrise and sunset are sufficient to synchronize farmer life. It all just depends on the individual culture.
Clocks have a deep and long history on earth. Sundials were used by ancient egyptians over 3000 years ago, and whilst sundials are deceptively simple, they are a very accurate way to tell time. The list goes on and on, candle clocks, hourglasses, astronomical clocks (which the sundial is technical a part of)...If there's one thing humans figured out pretty good in ancient times it was how to quantify and tell time.
Since you mentioned that this world is very Earth-like, I don't think it's likely for time systems to develop that are not anchored by the positions of celestial objects, as such systems do not appear common on Earth (I'm unaware of any).
An astrolabe typically manifests as a small tube with a protractor attached to it and a weight hanging from it. By aiming the tube at the sun (don't look through it!) or a star, you can use the hanging weight to find that object's altitude. And if your system of timekeeping happens to be based on that object's altitude, then you know the time, too (you may also need to know the day of the year since a star's path through the sky depends on that).
Time would probably be defined in terms of the sun during the day, and a particularly bright star during the night, preferably one that is distant from the celestial poles since circumpolar stars and other stars near to a pole don't appear to move as much. However, the stars that do move don't just move throughout the night, but also throughout the year, such that you may only see a particular star during autumn and winter but not during spring and summer.
Of course, this is a fantasy world, so we wouldn't have degrees on our protractor. We would have zodiac symbols or whatnot, and the position of the hanging weight would tell us that we are halfway through the hour of the lion.
In a city, it is likely that bells would be used to help people keep track of time. Since this is a particularly punctual society, there might be one bell to chime the hour, and another to chime how many periods of perhaps 5 minutes have passed since the start of that hour. This would mean a lot of chiming, but no one would have an excuse for being late.
In the case of a city using bells to announce the time, they still of course need a way to tell the time, and this could also be done based on the position of stellar objects, or they might have some other mechanism that allows them to do so. This could be a giant hourglass, a pendulum, or a water clock. There are many possibilities there, though even these time telling means would likely need to be occasionally synced with the time as represented by the positions of celestial objects. Methods like this are especially important on cloudy nights when the stars are not visible.
There's also the 12 hour day option. When the sun rises it's 6am, noon is noon, sunset is 6pm. There are always 12 daylight hours, always 12 hours of darkness. The fact that the hours are of variable length doesn't matter, that's just how the day runs. It also doesn't matter that this measure of time only applies over a very small area, there's no form of long distance travel or communication over which mere hours matter.
The reason accurate timekeeping is a subject not mentioned in medieaval periods is that it just isn't a thing. Only daylight matters. Things happen, one after the last, from first light until it's too dark to see, it really doesn't matter what time it is (until you need to navigate the open oceans).
For the farmers and serfs working the land, the cycles of the sun dominated the day to day work, weeks for markets and festivals and the seasons for the major events like harrowing and planting, campaigning season (when you might get called up to fight a war, or protect your crops), harvest and winter (where a lot of the repair and preparation work happened).
In the towns, there was a bit more need for time keeping, as you might need to coordinate several people meeting at the same time to buy and sell stuff, plot against the king or make a private transaction of some sort. Even then, the daylight hours would probably be divided into quarters, i.e. sunup, midmorning, noon, mid afternoon, sunset.
In the most highly regulated parts of the society (not economy), timekeeping became much more important. Monasteries were the places where you would be most likely to find clocks of various sorts (water clocks, "candle clocks" or even primitive mechanical clocks) in the European Middle Ages in order to ensure the members and lay members attended the regular cycle of prayers and devotions in the Christian day. Monasteries and churches would also ring bells at set times to remind the townsfolk when it was time to pray, come to church and perform other spiritual tasks, although obviously not as often as inside the church or monastery itself.
Modern timekeeping as we understand it only became important as people began carrying out long distance travel (calculating latitude and longitude requires accurate astronomical instruments and clocks, hence the introduction of naval chronometers), and synchronizing schedules for railway traffic on land. This is far beyond the middle ages, naval chronometers needed large advancements in technology and didn't become possible until the 1700's, while the idea of standard time and time zones needed for railway scheduling didn't appear until the 1800's.
The university where I grew up had a carolin that rang Westminster chimes every hour from 7 a.m. to 10 p.m. As a boy growing up, the 8 p.m. was my signal to be home, later moved to 9 on warm summer evenings.
3a8082e126