This is a great question that I wish I could answer but can't. However, I did want to drop a related point of interest concerning a resurgence in cavalry and pack animal use (!) that spans a period from the late 1980s to the NATO presence in Afghanistan. The latter is obviously beyond the subreddit's limit on content past 1995, so mods, if it's inappropriate, I'll delete. However, I think it's kind of an interesting guide en miniature to how Western militaries have rethought and approached combat in the post-Cold War period.

It was the opinion of the U.S military in 1942 that horses had long since outlived their use on the battlefield, and they more or less forcibly retired John Knowles Herr, the army's last Chief of Cavalry. Herr was widely considered a fossil who advocated for the continued use of an animal that was clearly outclassed by its mechanical alternatives. However, he may have had the last laugh. While the ensuing Cold War saw the Soviet Union and West constantly jockeying for technological superiority, the growth of what's referred to as fourth-generation or asymmetric warfare (warning: PDF) afterwards has shrunk battlefields and often placed them in regions that are very hard on advanced technology. Horses have re-found a niche in this context, even if mostly for special-forces soldiers.

So how'd this happen? The above article is a very influential piece from 1989 that I enjoy linking because it's almost creepily prescient about the state of geopolitics more than 20 years later. It observed that:

• Interstate violence in the beginning of the post-Cold War period was declining significantly. In plain English, countries weren't fighting each other as much anymore. (This is even more true today.) Particularly in relation to the Soviet Union and U.S., the high-technology approach to warfare suddenly found itself without its customary enemy.
• Trouble was more likely to emerge from non-state actors like terrorist groups. (Again, even more true today!)
• Western militaries had such a massive arms, technology, and training advantage that there was no realistic way for a non-state actor to defeat them in conventional warfare. If you put the average terrorist or insurgent group off in the middle of some unpopulated locale to face off against a Western army, the former would be annihilated.
• Therefore, the incentive for such groups is to engage in what's known as asymmetric warfare. This attempts to minimize the advantages of a modern army by forcing it to fight in places where its technological superiority can't be brought to bear easily, in urban environments where it's easy for civilians to get caught in the crossfire, and in the media for propaganda and recruitment victories. Basically, you try to direct conflicts to areas where it's difficult, expensive, or dangerous for a well-equipped Western military to use all its cool toys, or you make them fight in populated areas where the risk of collateral damage is unacceptably high with bombs or missiles, and you try to turn public opinion against them with civilian casualties or philosophical/cultural pressure. A terrorist or insurgent group still won't necessarily win under the circumstances, but the idea is to draw a conflict on long enough to create pressure on Western politicians and try to force a withdrawal.

Again, this was a very influential article that captured a lot of the discussion going on in the U.S. military while the Soviet Union was in its death throes. Shortly after it was published, the U.S. marines started to offer training courses for the use and care of mules/donkeys as pack animals in mountain/desert terrain, and there was increased emphasis placed on training for fights where you wouldn't necessarily have a "front," access to a stable supply line, or constant communication with superiors.

In 2001 with the NATO invasion of Afghanistan (we won't dwell on this too long as it's beyond the period of discussion for /r/AskHistorians, but I think it's worth a quick mention), it quickly became apparent that horses were really the only option for transport in rough terrain while supporting Northern Alliance attacks on the Taliban. Calling in a helicopter: a). would draw unwanted attention to yourself, and: b). was pointless anyway because the fuel, maintenance, and personnel structure in place to support them didn't exist. There's a book by Doug Stanton about this named Horse Soldiers that covers the American special forces' abrupt reintroduction to fighting from horseback, and how they wound up reinstituting training for it.

It would be the RKKA (Soviet Ground Forces) or the Royal Italian army. By 1942, most other countries had their cavalry on the downturn; they were either mechanizing, or being shipped off to different units. The US 1st Cav was unhorsed in 1942 and used as an infantry division, the US 2nd Cav, which had segregated units, never left stateside. All independent US Cavalry regiments were fully mechanized by 1942, or were in transition with 1 battalion horse, 1 battalion mechanized; these formations would be brought up to full mechanization upon deployment to Great Britain.

The British 1st Cavalry Division conducted its last campaign 'in saddle' when Commonwealth forces crushed Vichy Syria; afterwards the division would become the 10th Armoured, and would go on to fight in the North African campaigns. This trend mirrors other Cavalry divisions of the time; like the Wehrmacht's 1st Cavalry Division, which would become the 24.PanzerDivision in 1941-42.

As stated however, while other nations modernized their cavalry, the Soviets began to expand it; with entire Cavalry Corps and Cavalry Divisions (the latter in reality being about 3 battalions in size) often operating with Mechanized Corps. The 1st Guards Cavalry for example performed very well during Operation Uranus. RKKA Cavalry continued to do official border patrol duty until well into the 1970s from the saddle.

Likewise, less out of doctrine and more out of industrial considerations, the Italian army kept a few Cavalry regiments in service throughout their time on the Eastern Front. The last successful, massed cavalry charge is often credited to the Savoia regiment in 1942; when they managed to overrun a few Soviet infantry battalions in a hasty, bloody attack.

The Waffen SS retained Cavalry units for anti-partisan and frontline duty. The SS-Kavalrie-Brigade led by Fegelein would eventually expand into the 8.SS Florian Geyer, and saw a lot of action in Army Group Center and South's areas of responsibility, as well as anti partisan duty in Yugoslavia and the Prippet marshes. The Hungarians would also help form the 22.SS Kavalrie Divison "Maria Theresa." Together both Divisions would fight in the Budapest pocket, often as infantry, but 'from the saddle' as mounted rifles when fodder and care could be provided for their horses.

None of them, really. Most Commonwealth countries have cavalry units which still train on horses (even if for only ceremonial purposes) and Special Forces still use horses for transportation as the situation requires.

I was under the impression that the last major cavalry charge was the charge of the Savoia Cavalleria at Isbuscenskij.

August 1942 between the Italian cavalry and USSR. I'm sorry for not having any first hand sources, I just remember this being cited by my professor as an example of Italy's reluctance to use modern weapons and tactics.