After (and during) Overshoot there is likely to appear fundamental differences in the military equipment and tactics, due to fundamental differences in technology, between various countries, continents (more so than currently) and even within single militaries. It will however be different than when Europe almost conquered the world because most militaries will be aware of each others capabilities and it won't be the same sort of mismatch (especially in terms of military thought and organization/motivation) nor as big technological differences (probably) that was seen during the era of colonization, (importantly social technologies, such as Nationalism, will be fairly evenly spread throughout the world). I should also stress that it isn't similar to the situations faced by say Rome and most empires, that difference was normally either one of organization or of a slight technological edge combined with an efficient military. Rome in the beginning had either equal or inferior technology than its opponents, the Celts were superior iron workers, while the Carthaginian soldiers were roughly as effective as the Roman legions they faced. Parthia had better cavalry, while Rome had better infantry and so the terrain decided the border for them. What made the Roman armies superior was a combination of strategy (it was actually an extremely cautious one), training and organization (their communication system was far superior to the barbarians for example). The difference here is on the level of different fundamental technologies, such as in metallurgy, precise machining and so on, rather than the soldiers having different products of the same underlying technology (such as the wide variety of assault rifles currently available).
An example of the differences could be one area which cannot mass produce automatics and so either imports them from an area which can mass produce automatics or has small scale artisanal production with the majority of the military armed with bolt action rifles. Or a specific area cannot create decent guns and so uses bows or arbalests (advanced crossbows) and neighbours a region which can produce non-automatic rifles. That isn't as lopsided as it sounds, at one point it was seriously considered to replace WW1 era guns with longbows because a well trained long bowman has significant tactical advantages over a WW1 soldier. Logistics (it would have taken a lot of trees) and training ("to get a great longbow man, start with his grandfather", old saying) were the main problems with the idea, not the ineffectiveness of longbows. This sort of technological imbalance will most obviously affect ships, vehicles, aircraft and artillery, drastically affecting the tactical and strategic landscapes, infantry due to their ability to use terrain/cover the best and their low cost gear will likely be affected the least. These technological differences will exist because of fundamental differences in resources, social organization and supporting technology available across the world.
These differences will also affect each branch differently. The air forces will likely be the most affected and could quite easily disappear from large areas, or return to pre-WW1 forms with recon balloons. Dogfights in WW1 were often over these balloons and with airships likely to have a greater role, this role of aircraft could see a resurgence albiet one that is more mobile (like mortal engines). Where they stay, they'll probably be reduced to attachments to the army or navy, rather than as a separate organization and be focused on recon, emergency supplies and critical strikes rather than today's use of them for massed bombing and close air support. If airships appear then they'll be mostly transport and recon, or even mobile airbases (unlikely but possible). More advanced air forces will have a major advantage over more primitive air forces, so much that significantly inferior ones could only be used opportunistically or by buying advanced planes from other states, though it should be noted that if the tech gap is small then enough numbers on the inferior side could outweigh that advantage somewhat. Fueling them will be the major challenge and chances are that if they're not needed, they won't be used simply to preserve fuel and only major wars or battles will see them. This would mean that overall pilot quality would be much lower due to less training flights, though it is possible that many militaries would have several squadrons of elite pilots who train regularly. This also means that most skirmishes or battles would not see aircraft.
The army is different from both the navy and air force in that while technological differences matter, they aren't nearly as decisive. Think of the Vietnam war, where the technologically inferior Vietcong defeated the technologically superior USA. To quote Napoleon (and he knew what he was talking about)“In war, the moral is to the material as three is to one.”, other factors like moral, discipline, organization, leadership, numbers and such have a far greater impact than technology on its own. Generally though, the armies that have access to the best technology also happen to have good morale, training, leadership and such (That this was not true in Vietnam was one of the reasons for a US defeat, and something similar is playing out in Afghanistan). When these qualities are lacking, no amount of gadgets or equipment can make up for them. This is why Pakistan's and India's armies are walking farces (compared to western and east Asian armies at least), along with many African armies (how many rebels does it take to beat them again). As Machiavelli said, "I maintain, therefore, that it is not gold, as is vulgarly supposed, that is the sinews of war, but good soldiers".
Technology however has an effect on what qualities are important and how they manifest. Courage is a good example, is it courageous to hold your fire until the right moment, fight in single combat or not leave the formation. The answer depends on what warfare type and weapons are in play (respectively, Hezbollah, Tribal and Spartan). Some values are suppressed or favored, initiative was vital for the early successes of Germany during WW2 but when fighting in large blocks with muskets its not needed and can be counter-productive in low level officers. Additionally armies are likely to have a wide variety of technology, elite squads with automatic weapons, massed conscripts with rifles, militia with bows or arbalasts fighting alongside each other. Or velomobile towed artillery or machine guns working alongside their horse drawn cousins. This will by its nature make militaries both more complex and more resilient, except if its badly managed in which case they would be very brittle. The military cultures generated by these differences could vary by a significant amount, even inside the same military, which would be a source of friction (especially if that means there are several military organizations of varying sizes and capabilities like in Iran).
Also Guerrilla warfare is, while it will still be around, likely to both diminish and increase in importance (this is due to the return of conventional wars as common across most of the world and the likelihood of many such conflicts springing up as central governments lose control over their peripheries). There are a couple of reasons for this, to quote Col. David H. Hackworth (110 medals and 25 years of combat experience in Korea and Vietnam and author of About Face) "Guerrilla war, not push button war, was the wave of the future - it could only flourish under the protection of the bomb" and that protection will disappear. As this describes, escalation is limited when nuclear war is a possibility and this stops many of the actions that can decisively end guerrilla war (such as bringing the war to a final conclusion), while also keeping some wars in a low enough intensity that it allows it to continue, (though there has been advances in many social technologies that have also contributed). Also since an army has to be able to fight the highest intensity threat around, modern militaries are taking a while to adapt to low intensity warfare. To give an example, I remember reading about a new technique/division that was tried near the end of the Vietnam war, it wiped out the Vietcong in the area by simply killing so many of them (it was a special forces operation) that no replacements existed, there are a couple of other tricks (most however are vary from morally grey to black). The protection civilians currently have in war is likely to disappear, so rape of Nanking style events will likely become more common (note the countryside isn't immune and these events will likely be at worst uncommon across most of the world), this can destroy the civilian support guerrillas receive and has further implications.
Terrain also has a massive impact on the choice of technologies and how effective they are. Its not an accident that Europe throughout history fielded primarily infantry armies while in the middle-east cavalry was far more common (in fact knights were invented in the Middle East). The border between Parthia and Rome was formed around this, on Parthia's side it was flat and favored cavalry, while on the Roman side it was hilly and favored infantry. No horse army has conquered Europe (despite coming close) because of its terrain and in the future terrain will play a much larger role in the technology chosen than it does now. So some areas won't have certain vehicles, simply because the terrain disadvantages them (heavy tanks are likely to see this often, especially due to their high fuel use). Logistics also plays into it, both from what support vehicles can be used and how difficult transport is. Since a lot of advancements happened in the Napoleonic ear, such as improved carts, its not likely for military logistics to degrade hugely,but degrade they will. Besides ancient empires fielded armies of significant size before, its just that gun based armies require far more support than sword based armies.
This greatly affects strategy, tactics and training. What works against one technology suite doesn't against another. So imagine a state which has two technologically different neighbours and how their military copes (say a small, highly mechanized force and a large bolt action rifle and artillery force). Or an army which is used to fighting one technological suite suddenly fighting a different one. How about mercenaries that work across continents (like the Swiss did), how are they supplied and how do they deal with the varying technologies.