The most important calendar in early thirteenth-century Khwarazmian Uzbekistan* would have been the Islamic or hijrī calendar. By this time much of Central Asia had at least nominally converted to Islam, with some of the major cities—Bukhara in particular—boasting a centuries-old religious tradition. The Khwārazmshāh himself had been Muslim since the late eighth/early ninth century, though in this respect the ruler was far in advance of the much more slowly converted populace, and one—the late twelfth-century ʿAlāʾ al-dīn Tekish—had been confirmed as ruler of his vast domain by the ʿAbbasid caliph. It is therefore unsurprising that the Muslim rulers and urban population of the region would have kept to the calendar of their faith.
The early thirteenth century, however, saw something of a shakeup in these demographics. The Mongol conquest of Khwarazm in 1219-1221 removed the old leadership from power and replaced it with a new, notably non-Muslim, group with a new, notably non-Muslim, calendar. The Mongol calendar, like the Chinese, relies on a cycle of twelve years named after twelve totemic animals; each year consists of twelve lunar months, with the periodic insertion of a thirteenth intercalary month to keep the calendar in alignment with the solar year. Sometimes, however, the months were grouped into seasons rather than referred to individually; thus, for example, the first month of the calendar might be referred to as "Aram [ay]" or "the first month of Spring." (The extant evidence suggests that chronicles tend to refer to months by name and official chancery documents tend to refer to months by season, but the reason for this divergence remains unclear.)
Of course, those Mongol states which ruled over substantial Muslim populations neither could nor wanted to entirely abandon the Hijrī calendar; in many cases, their bureaucracies drew heavily on the scribal classes which had performed such tasks before the Mongol conquest. Thus, the Golden Horde, the Ilkhanate, and the Chaghatay Khanate (this last being the one which controlled modern Uzbekistan, and is thus most relevant to your question) continued to use the Hijrī calendar, a preference which aligned neatly with the ruling class's conversion to Islam in all three cases. Chaghatayid coinage, for example, was generally struck with Arabic legends and Hijrī dates from the time of Chaghatay (who, I should stress, was not himself a Muslim) onwards; there are some coins which do not bear dates, but none I am aware of that hew to an alternative system. At the same time, the traditional Mongol animal-and-season calendar was retained, especially in official decrees and other documents—perhaps to retain some degree of uniformity with the Yuan dynasty in China, which had no reason to officially adopt the Hijrī calendar and was theoretically preeminent among the successors of the unified Mongol Empire.
Other calendars would have been known to the thirteenth-century inhabitants of Uzbekistan, but used rather less often. We can get some idea of the calendars considered relevant in the eastern Islamic world (if not Uzbekistan specifically) through the genre of ʾazyāj (sing. zīj), texts consisting of tables and algorithms for calculating the positions of astronomical bodies at different calendar dates. The three most relevant of these are the Zīj al-Sanjarī, composed around 1115 CE in Marv (modern Turkmenistan) by al-Khāzinī; the Zīj-i ilkhānī, composed around 1271 by Nasīr al-dīn Ṭūsī from data gathered at the Ilkhanid observatory at Marāgha (NW Iran); and the Zīj-i sulṭānī or Zīj-i Ūlūgh Beg, composed around 1439 by a group of astronomers in the employ of the Timurid prince Ulugh Beg at his observatory in Samarqand. While some of the calendars mentioned in these works would have been of negligible value to anyone outside of a formal observatory (some, like Khāzinī's "calendar of Bīlibbus," which reckons time from the ascension of Philip III of Macedon, were probably useless even to astronomers), one type of calendar mentioned in all three texts is perhaps worthy of further remarks.
This is the Zoroastrian solar calendar and its derivatives, which were important, among other things, for establishing the date of the Nowruz festival. (Here's the catch: while Nowruz and other festivals of Zoroastrian origin were certainly celebrated in medieval Iran and early modern/modern Uzbekistan, I can't say with certainty that they were celebrated in thirteenth-century Uzbekistan. If not, these solar calendars would possibly have been of more limited use.) This calendar was reformed on the orders the Saljuq sultan Jalāl al-dīn Mālik-Shāh I in 1073 (and first implemented in 1079) as the "Jalālī calendar," which in turn was the basis of the "Khānī calendar" instituted by Maḥmūd Ghāzān Khān in Iran in 1302. Transoxianan astronomers would have been well aware of these developments, but there's no evidence to suggest that the Zoroastrian or Jalālī year (much less the Khānī year, which was much more obviously restricted to the Ilkhanate) played much of a role in everyday life in the region.
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Further reading:
Abdollahy, Reza. "Calendars. II. In the Islamic Period." Encyclopedia Iranica. https://www.iranicaonline.org/articles/calendars
Melville, Charles. “The Chinese-Uighur Animal Calendar in Persian Historiography of the Mongol Period.” Iran 32 (1994): 83–98. https://doi.org/10.2307/4299907. [open access]
Thomann, J. (2021). "Chapter 5 The Institution of the Jalālī Calendar in 1079 CE and Its Cohabitation with the Older Persian Calendar". In Calendars in the Making. Leiden, The Netherlands: Brill. doi: https://doi.org/10.1163/9789004459694_007. [open access]
Montelle, Clemency. “The ‘Well-Known Calendars’: Al-Khāzinī’s Description of Significant Chronological Systems for Medieval Mathematical Astronomy in Arabic.” In Calendars and Years II: Astronomy and Time in the Ancient and Medieval World, edited by John M. Steele, 107–26. Oxbow Books, 2011. http://www.jstor.org/stable/j.ctt1cfr922.8.
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* To clear up any misunderstandings here, I think I should note that Transoxiana—or its Arabic equivalent, Mā warāʾ al-nahr/Mawarannahr—is a geographic rather than historical term, referring to the area between the Oxus (=Amu Darya) and Jaxartes (=Syr Darya) rivers. So much of thirteenth-century Uzbekistan (though not all—the site of modern Tashkent, for example, lies beyond the Syr Darya) was part of Transoxiana, but it was also part of Transoxiana before the thirteenth century and remains so to this day.
By contrast, while Khwarazm is also a geographic term, its usage in that sense only includes a relatively small part of modern Uzbekistan centered on the administrative region of Xorazm. The medieval state of Khwarazm, however, did encompass pretty much all of modern Uzbekistan until the Mongol conquest in the early thirteenth century. So I'm using it here as a political/historical rather than geographic label.