While it would be definitely better to wait for the answer from proper specialists, I, though as a non-specialist in this area, by change know the relevant text.

Ruy González de Clavijo (d. 1412), an Castillan diplomatic envoy from King Henry III of Castille (r. 1390-1406), travelled to Samarkand, capital of the Timurid Empire in the last period of Timur's reign (r. 1370-1405) in fall 1404, and records his itinerary as well as diverse things in Historia del gran Tamorlan e itinerario y enarracion del viage y relacion de la embaxada que Ruy Gonçalez de Clavijo le hizo (History of the Timurid Empire, itinerary, and its narrative, related by the ambassador, Ruy González de Clavijo). The text is published in original Spanish in the 16th century, and some translations are also available in other European languages including English.

In his description of the capital, Clavijo notes as following:

'Thus trade has always been fostered by Timur with the view of making his capital the noblest of cities: and during all his conquests wheresoever he came he carried off the best men of the population to people Samarqand, bringing thither together the master-craftsmen of all nations. Thus from Damascus he carried away with him all the weavers of that city, those who worked at the silk looms. Further the bow-makers who produce those cross-bows which are so famous: likewise armourers: also the craftsmen in glass and porcelain, who are known to be the best in all the world. From Turkey he had brought their gun-smiths who make the arquebus, and all men of other crafts wheresoever he found them, such as the silver-smiths and the masons. These all were in very great numbers, indeed so many had been brought together of craftsmen of all sorts that of every denomination and kind you might find many master-workmen established in the capital. Again he had gathered to settle here in Samarqand artillery men, both engineers and bombardiers, besides those who make the ropes by which these engines work. Lastly hemp and flax had been sown and grown for the purpose in the Samarqand lands, where never before this crop had been cultivated'.

English translation is taken from: Guy Le Strange (trans.), Clavijo. Embassy to Tamerlane 1403-1406, New York and London: Harper, 1928, found here.

While this description itself does not directly prove the Timurid use of firearm weapons in the actual battlefield like the famous battle of Ankara in 1402, the expansion of his dominion by military conquest certainly enables Timur to access this kind of weapons and at least he showed some interest in encouraging its product.