The answer to this question lies in a combination of two things: a certain level of specificity (i.e. details) and, by simple happenstance, why the Rus' had guns whilst the (Tartar)-Mongols did not.
On details: Specifically, the question is on guns (not fire-bombs, fire-lance, etc.) and Great Horde (not Golden Horde).
On timing: The standoff at Ugra River in 1480.
I will try to answer this: "Why then, many years later, did the Mongols lose access to firearms and gunpowder weapons, while their subject Russians gained them?" (last sentence by OP).
Development of the Modern Gun
First, the the modern gun was developed specifically around this period, late-15th century:
The classic handheld firearm emerged in Europe at the same time that classic artillery did — in the last decades of the fifteenth century. In illustrated chronicles of the 1480s, soldiers fire guns that look recognizably modern (Figure 12.1). They have long, thin barrels, and they are held close to the cheek, one eye peering down the barrel to aim. Although it’s not clear in Figure 12.1, these firearms had a lever mechanism that allowed a burning fuse to be lowered into the flash pan by means of a simple movement of the finger. This mechanism, known as a matchlock, was a significant advance because it enabled a soldier to hold the gun at eye level. With its butt resting against his shoulder, he could steady the gun with one hand and fire with the other. In the following decades, trigger mechanisms gained springs and other refinements, and the guns became even more convenient.
Having achieved this classic form, firearms began to appear more regularly on European battlefields. In the 1480s, gunners were still vastly outnumbered by bowmen, swordsmen, and pikemen, but their numbers were growing steadily. Spanish records show that the proportion of matchlock units to crossbow and bow- and- arrow units increased significantly in the late 1480s and early 1490s, a process driven by the constant experimentation of the Granada Wars (1481– 1492). Spanish gunners brought their new techniques to Italy during the devas- tating wars that started in 1494, to decisive effect, as in the famous 1503 Battle of Cerignola. Thereafter arquebusiers became increasingly prominent in Europe, so that by the late 1500s they had become a core component of European armies, reaching proportions of 40 percent of infantry forces.
Source: Andrade, Tonio (2016), The Gunpowder Age: China, Military Innovation, and the Rise of the West in World History, Princeton University Press, p.167.
A simpler way of putting this, Wikipedia's timeline on the gunpowder age states clearly, "(in) 1480 ... Guns reach their classic form in Europe". From here, we can presume Ivan the Great imported them for his army.
Allow me to digress and point out, with particular focus on Ivan's enemies on the Ugra River in 1480, the so-called Mongols ... again the details matter. The correct term should be Tartar-Mongols (also known as the Great Horde, and not to be confused with Golden Horde of the Jochids). The importance of this distinction will be explained in a moment.
Mid-15th century: Guns from Europe, not China
Second, one key element of the question "... (why) did the the Tartar-Mongols lose access to firearms ...?" The short answer is, they never had it in the first place, because guns (the instrument itself, not gunpowder) were developed in China and Europe, and it was improved dramatically in the latter (Europe) during mid-15th century (a few decades just before the standoff at Ugra River):
In fact, guns in China were developing along a similar trend to those of Europe, growing longer relative to muzzle bore. But the development slowed in China about a generation before the development in Europe of the classic cannon. Why? The reason probably has less to do with to do any putative cultural ingenuity on the part of Europeans than with the frequency of warfare. After 1449, China entered a period of relative peace, while Europe entered a period of sustained, intense, existential warfare. By existential warfare I mean conflict that threatened the very existence of the states involved. Chinese guns had evolved quickly between the late 1200s, when the first true guns seem to have emerged, and the early 1400s, a period during which China was wracked by existential warfare. The century from 1350 or so to 1449 was especially turbulent, as the Ming strove to establish and consolidate their empire, and during this time the evolution of guns, toward longer barrels, seems to have been proceeding along quite similar lines in China and the West. In the middle of the 1400s, this evolution stopped in China and accelerated in Europe, precisely when warfare decreased in China increased in Europe.
Source: ibid., p.105.
Great Horde (non-Chinggisid, therefore no support from other Mongols)
Finally, the important distinction of the Great Horde, as opposed to the Golden Horde, comes down to this, the Great Horde was merely one of many different 'hordes' after the disintegration of the Golden Horde in late-14th century. Most important, they were not related to Genghis Khan (i.e. non-Chinggisid).
A short paragraph on the many subsequent hordes (or khanates), post-Golden Horde:
With Toqtamish’s overthrow in 1395, a new clan, the Manghit (see quote below), under the non-Chinggisid commander in chief Edigü (d. 1420), emerged between the Volga and the Emba. Edigü maintained something of the Horde’s unity until 1411, but by 1425 independent regimes were ensconced throughout the Golden Horde’s territory. Khanates of Blue Horde origin formally pro- claimed themselves in Crimea (1449), Kazan’ (or Bulghar al-Jedid “New Bulghar,” 1445), and Kasimov (1453). The Crimean khanate finally dispersed the “Great Horde” (Ulugh Orda), composed of the right-hand Sanchi’ud (Turkish Sijuvut) clan, in 1503.
Source: Christopher P. Atwood, "Encyclopedia of Mongolia and the Mongol Empire" NY: Facts On File, Inc, 2004, p.208.
The Mangghud, Manghud (Mongolian: Мангуд, Mangud) were a Mongol tribe of the Urud-Manghud federation. They established the Nogai Horde in the 14th century and the Manghit Dynasty to rule the Emirate of Bukhara in 1785. They took the Islamic title of Emir instead of the title of Khan since they were not descendants of Genghis Khan and rather based their legitimacy to rule on Islam.
To end, even if the East Asian Mongols wanted to help provide guns to the Great Horde, it would have been a stretch simply because, by this stage, China was no more under the Yuan dynasty, it had shifted to the Ming (1368–1644).