I'm not quite sure what changes you're referring to. The long-haul desertification of the Sahara took places millennia and millennia ago; Timbuktu didn't coalesce into a settlement and trade center until probably the eleventh century CE, and it didn't become the political center of the Mali Empire until the fourteenth. (It's unclear where the earlier capital was--locations in modern Guinea, on the Gambia River, and in the vicinity of Bamako, Mali have been suggested by scholars based on the writings of medieval Arab writings).

Pitched on the southern edge of the Sahara, in the north-ish and eastern part of the Mali Empire, Timbuktu was well and truly a desert by the late Middle Ages. The very first thing that al-Umari (writing in Cairo, on the basis of reports from people who had crossed the Sahara to Mali and returned) says about the Mali Empire, ca. 1335, is:

The country is very hot.

One of the only direct references to the climate of Timbuktu, specifically, in medieval Arabic sources is explicit. Al-Maqqari's 17th century text is a compendium of reports from mostly-lost earlier sources, and from a writer we date to the 14th century, al-Maqqari refers to Timbuktu as

a place in the desert, which is one of the provinces of Mali.

Leo Africanus (al-Hasan ibn Muhammad al-Wazzan al-Fasi or al-Zayyati) in the sixteenth century is more evocative about the human ecology of Timbuktu, that is, the interaction of humans with the environment:

There are many wells containing sweet water in Timbuktu; and in addition, when the Niger is in flood canals deliver the water to the city...There are no gardens or orchards in the area surrounding Timbuktu.

The city is very much endangered by fire. At the time when I was there on my second voyage, half the city burned in the space of five hours. But the wind was violent and the inhabitants of the other half of the city began to move their belongings for fear that the other half would burn.

That said, it's importance to recongize the long-term (75-100) year climate cycles that characterized Saharan conditions during the Middle Ages and early modern era, alternating periods of even more heat and aridity with (comparatively) cooler and wetter periods. The difference was enough to, in the hotter periods, dry up wells and force long-haul traders to circumnavigate the Saharan to the south or to the north and then west rather than crossing it between Mali and Egypt. The drier periods likewise hampered both agriculture and pastoral activity among settled peoples on the fringes of the desert, resulting in increased nomadic activity and warfare. Thanks to its position on the Niger, as referenced above, Timbuktu endured as a thriving city during the drier periods. Indeed, it thrived--the redirection rather than extinction of trade routes to the Maghreb and to Egypt is proof of that. Nevertheless, one imagines that the city would have felt some of the changes especially thanks to its starring role in trade. The wetter periods, in fact, facilitated the gold mining that was such a valuable export for West Africa during these centuries, when the rain would wash down ore-heavy gravel from mountains/hills into the Senegal and Niger rivers. So between slightly cooler temperatures, increased gold traffic from the west and much happier caravan travels from the north and east, Timbuktu probably enjoyed those comparatively wet periods quite nicely.