Why did Russians have guns towards the end of the Mongol Yoke, but not the Mongols?

Here is a technology-based answer. I admit that I don't know the historical reason why the Russians used guns and the Tatars did not, however, I can provide reasons why it would make much more sense for the Russians to use them than the Tatars.

Guns, before the widespread adoption of flintlock in the late 17th century, were close to useless in the hands of cavalry.

Guns before the 16th-17th centuries couldn't be just taken out of a holster or saddle-bag and fired at the enemy. They had no priming mechanism. You had to ignite them with a burning piece of rope called a match cord. Take a look at matchlock muskets: Even if the gun was already loaded, you had to insert the match cord into the serpentine, adjust the position of the burning match so that it would strike the pan (as it burned it got shorter so you had to keep re-adjusting it), then open the pan, then blow on the smoldering matchcord to make it light up, and only then could you pull the trigger.

https://www.youtube.com/watch?v=2KTS8PQ06Qo

And if we are talking about even earlier times, like 14th and 15th century, it was even worse, because you didn't even have a trigger: you had the gun in one hand, burning matchcord in the other hand, and you touched the match to the ignition hole manually. Good luck doing that while riding a horse! (Even on foot it was close to impossible to aim while holding the gun in one hand and match cord in the other, so often two men were used, one to aim and the other to ignite the gun) Even reloading would be troublesome, on horseback where would you put that burning piece of rope while you reloaded? Also lighting a fire wasn't easy back then, especially while galloping. Often both ends of the match cord were on fire, because firing the gun might extinguish the end you used.

Infantry had no such problems, they could even rest the musket on supports.

Note: there were guns optimized for cavalry, the wheellock. However, it was a very complicated, expensive and maintenance-intensive piece of equipment, suitable for knights, not so for steppe nomads. And it wasn't invented yet in our timeframe.

So, to sum it up: guns in that time period were only suitable for infantry, not for cavalry. The Tatars used almost exclusively cavalry. It's much easier for an army already composed mostly of infantry to adopt a weapon well suited for infantry, than for an army and culture composed mainly of horseman to abandon horses and become infantry.

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.

Source: Wikipedia

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).

First of all I agree with Semaphore's first comment: it is unlikely that in 1480 firearms could make much difference. Second, the adversary of the Russians in 1480 were not Mongols. They were called Tatars, and they were remote descendants of the original Mongol empire. The technology of Mongols in the 13th century was based on Chinese technology. By the 15th century, Tatars who fought with the Russians had little connection with China, and if there is any connection between them and the original Mongols it is mostly historical. And who can tell for sure that Tatars did not have firearms in 1480?

With all this a broader question remains. It is a common opinion of historians that gunpowder, firearms and rocketry were invented in China. Where did all this disappear? When the Europeans came into direct contact with China, and other countries in the far East, they hardly mention any Chinese artillery and rockets. Probably they were not too impressive, even if they existed.

What existed from the older times was probably not very effective, and it did not develop. One can ask why it did not develop, but perhaps a more reasonable question would be why everything developed so fast in Europe.

EDIT. Regression in technology actually happens, and there are many examples. "Greek fire" was a terrifying weapon if we believe the contemporary descriptions, but we hear nothing of it when the Turks besiege Constantinople. It is lost and we even do not know what it was. Let me also mention the huge regression in technology which happened in Europe in the late Antiquity. Medieval (spring) artillery was inferior to Hellenistic artillery. Roman warships were (probably) inferior to Hellenistic warships (if we believe their contemporary descriptions). But it also happened in other times and places.

The issue is not that Russian "guns" gave them the advantage over the Mongols (or Tartars). The issue is that Russian guns equalized them with Mongol bowmen.

The Mongols (and English) were able to field armies with 60% bowmen because these bowmen were subjected to "continual" practice over a period of years. No other medieval armies trained such large percentages, or even large absolute numbers of men who could use missile weapons.

Guns, unlike bows, were something that "average" soldiers could learn to fire in a matter of days, or weeks. Now the Russians could also arm their troops with "range" weapons. This, in turn, neutralized the speed of the Mongolian horses.

Because they were fighting on their home ground, and enjoyed larger numbers, the Russians enjoyed "draw odds" (a draw counts as a win).

Put another way, "guns" represented at most a small upgrade over existing Mongol weaponry, but it represented a vast upgrade over Russian swords and spears, giving them the "equality" they needed to beat the Mongols, given superior numbers.