In the late middle ages it became common to install large mechanical clocks in towers at the town halls of cities and towns in Europe, to display the time to everyone who looked at them.
Few private citizens would have had their own clocks, and pocket sized clocks - watches - were only invented in the very late middle ages. So the few people who did have their own clocks could check the time from the more accurate public clock and reset their clocks.
And the displays of public clocks were often highly elaborate. For example, an automation might come out and strike a bell every hour, striking a different number of times each hour.
According to Wikipedia:
These mechanical clocks were intended for two main purposes: for signalling and notification (e.g. the timing of services and public events), and for modeling the solar system. The former purpose is administrative, the latter arises naturally given the scholarly interests in astronomy, science, astrology, and how these subjects integrated with the religious philosophy of the time. The astrolabe was used both by astronomers and astrologers, and it was natural to apply a clockwork drive to the rotating plate to produce a working model of the solar system.
Simple clocks intended mainly for notification were installed in towers, and did not always require faces or hands. They would have announced the canonical hours or intervals between set times of prayer. Canonical hours varied in length as the times of sunrise and sunset shifted. The more sophisticated astronomical clocks would have had moving dials or hands, and would have shown the time in various time systems, including Italian hours, canonical hours, and time as measured by astronomers at the time. Both styles of clock started acquiring extravagant features such as automata.
In 1283, a large clock was installed at Dunstable Priory; its location above the rood screen suggests that it was not a water clock. In 1292, Canterbury Cathedral installed a 'great horloge'. Over the next 30 years there are mentions of clocks at a number of ecclesiastical institutions in England, Italy, and France. In 1322, a new clock was installed in Norwich, an expensive replacement for an earlier clock installed in 1273. This had a large (2 metre) astronomical dial with automata and bells. The costs of the installation included the full-time employment of two clockkeepers for two years.
A sophisticated water-powered astronomical clock was built by Al-Jazari in 1206. This castle clock was a complex device that was about 11 feet (3.4 m) high, and had multiple functions alongside timekeeping. It included a display of the zodiac and the solar and lunar paths, and a pointer in the shape of the crescent moon which travelled across the top of a gateway, moved by a hidden cart and causing doors to open, each revealing a mannequin, every hour. It was possible to reset the length of day and night in order to account for the changing lengths of day and night throughout the year. This clock also featured a number of automata including falcons and musicians who automatically played music when moved by levers operated by a hidden camshaft attached to a water wheel.
In Europe, there were the clocks constructed by Richard of Wallingford in St Albans by 1336, and by Giovanni de Dondi in Padua from 1348 to 1364. They no longer exist, but detailed descriptions of their design and construction survive, and modern reproductions have been made. They illustrate how quickly the theory of the mechanical clock had been translated into practical constructions, and also that one of the many impulses to their development had been the desire of astronomers to investigate celestial phenomena.
Wallingford's clock had a large astrolabe-type dial, showing the sun, the moon's age, phase, and node, a star map, and possibly the planets. In addition, it had a wheel of fortune and an indicator of the state of the tide at London Bridge. Bells rang every hour, the number of strokes indicating the time. Dondi's clock was a seven-sided construction, 1 metre high, with dials showing the time of day, including minutes, the motions of all the known planets, an automatic calendar of fixed and movable feasts, and an eclipse prediction hand rotating once every 18 years. It is not known how accurate or reliable these clocks would have been. They were probably adjusted manually every day to compensate for errors caused by wear and imprecise manufacture. Water clocks are sometimes still used today, and can be examined in places such as ancient castles and museums. The Salisbury Cathedral clock, built in 1386, is considered to be the world's oldest surviving mechanical clock that strikes the hours.
It is possible that wizards study astronomy due to a belief in astrology and/or a belief that the positions of astronomical bodies influenced the success of magic, so that it would be good to time the castng of an important spell to when "the stars are right" for that particular spell.
Ancient people considered the planets to be the five dots of light that moved relative to the background stars - Mercury, Venus, Mars, Jupiter, and Saturn, and often added two discs that were seen in the sky, the Sun and the Moon.
Astronomers who accepted the heliocentric theory of Copernicus took the Sun and the Moon off the list, and added the Earth, mking 6 plaents. Uranus, Neptune, and Pluto were added to the last, raising the total to 9, and then Pluto was reclassified as a dwarf planet, reducing the total to 8.
In the first half of the 19th century the first four asteroids were often counted as planets, raising the total to 11, 12 after Neptune was discovered, and sometimes the first 15 asteroids were all counted as planets, bringing the total to 23 before all the asteroids were reclassifed as non planets.
At the present time five bodies in the solar system are classified as dwarf planets Pluto, Ceres, Eris, Haumea, and Makemake, and other bodies may become classified as dwarf planets in the future. So there are 8 planets and five dwarf planets, or 13 in total, and if Earth isn't included it would make 12.
The existence of as yet undiscovered planets in the far outer reaches of our solar system is considered possible.
At the present time astronomers are considering the evidence for the possible existence of a planet which is nicknamed "Planet Nine". Planet Nine might have a mass of about 5 to 10 times Earth, and orbit 400 to 800 Astronimical Units from the Sun.
There was the theory of a planet Vulcan inside the orbit of Mercury:
Theia, a planet which may have collided with Earth and formed the Moon:
A Counter-Earth, orbiting directly opposite Earth and thus hidden by the Sun. Muggle astronomers believe such a world is impossible, but maybe wizards believe it exists through magic:
And others. See the list of hypothetical solar system objects:
And it is possible that wizard astronomers used magic to discover planets which muggle astronomers have not yet discovered.
Modern muggle astrologers sometimes consider the influence of planets undiscovered by science.
Some astrologers have hypothesized about the existence of unseen or undiscovered planets. In 1918, astrologer Sepharial proposed the existence of Earth's "Dark Moon" Lilith, and since then, some astrologers have been using it in their charts; though the same name is also (and now, more commonly) used in astrology to refer to the axis of the actual Moon's orbit. The 20th-century German school of astrology known as Uranian astrology also claimed that many undiscovered planets existed beyond the orbit of Neptune, giving them names such as Cupido, Hades, Zeus, Kronos, Apollon, Admetos, Vulcanus, and Poseidon, and charting their supposed orbits. These orbits have not coincided, however, with more recent discoveries by astronomers of objects beyond Neptune.
Other astrologers have focused on the theory that in time, all twelve signs of the zodiac will each have their own ruler, so that another two planets have yet to be discovered; namely the "true" rulers of Taurus and Virgo. The names of the planets mentioned in this regard by some are Vulcan (ruler of Virgo) and Apollo, the Roman god of the Sun (ruler of Taurus). Another version of this theory states that the modern planets discovered so far correspond to the elements known to the ancients—air (Uranus, god of the heavens), water (Neptune, god of the sea), and fire (Pluto, god of the underworld)—which leaves the elements earth and ether (the fifth element of the fiery upper air). In other words, it is claimed that the two planets to be discovered will be named after an earth god or goddess (such as the Horae), and after Aether, the Roman and Greek god of the upper air and stars.
So we can wonder whether wizards count the Earth as a planet for all purposes. If they are considering the astrological influence of other planet on Earth, it would be illogical to count Earth, so that would make 12 planets on the watch not counting Earth, or 13 counting Earth. But maybe Earth is one of the 12 on the watch.
Do wizards count the Sun and the Moon as planets, as Muggle astrologers do? If so, there would be 10 planets among the 12 wizard planets, as opposed to the 8 planets of modern astronomy.
So if wizards count the Sun and the Moon as planets for the purpose of Dumbledore's watch, they would have to believe in two other planets unknown to muggle astronomers if wizards count Earth as a planet, or three other planets unknown to muggle astronomers if wizards don't count Earth as a planet.
So if wizards don't count the Sun and the Moon as planets for the purpose of Dumbledore's watch, they would have to believe in four other planets unknown to muggle astronomers if wizards count Earth as a planet, or five other planets unknown to muggle astronomers if wizards don't count Earth as a planet.
So as you can see there are many possible combinations of known, possible, scientifically disproved, theoretical, and more or less mythical or imaginary or mystical, solar system objects whichh wizards might possibly consider planets and count as the 12 planets on Dumbledore's watch.