The speculation about Egypt and Sumeria is certainly exciting; if we grant that the discovery/invention of prototypical gunpowder is low-probability and dependent on there being an extraneous tradition of alchemy to provide for a sufficient number of people fooling around with mixtures of the appropriate substances (charcoal or something similar to burn; sulfur, and the right kinds of nitrates for accelerant and oxygen supplements) then in the abstract, the odds of any one particular society in any one particular time chancing upon the early forms, and then developing them to the point of usefulness in either firearms or rockets, is quite low. But on the other hand we can't rule out the possibility of it being stumbled upon by any society with access to the basic ingredients either. Egypt has lots of depth of time to play with, and so does Sumeria. Both developed early civilization to a level where they were trading over a wide area (or rather, traders would come to them, especially in the case of Egypt) so they'd have a fair chance of developing it long before the Chinese stumbled upon it OTL, opening up speculations on the possibilities of Bronze Age firearms.
I forget now whether other threads on this sort of thing pointed out other technological limits that would keep guns marginal until say metallurgy developed to a level allowing gun barrels that can deliver a useful charge of shot. Offhand, I'd think that long before ironworking became common, copper-based metallurgy could deliver adequately strong alloys--but the raw materials would be rare, meaning guns would be feasible but not very common. Which would probably suit Bronze-age state builders just fine! It would take a while before iron alloys suitable for gun barrels could come along; we could easily then have an Iron Age where handheld iron and even steel edge weapons greatly outnumber the guns, yet the brass guns remain important. Indeed brass cannon retained significant advantages long after iron and steel ones also became common--much better resistance to corrosion for instance made them especially desirable for ship's guns and coastal defenses--a brass cannon could be fished out of a shipwreck and refurbished long after a ferrous one was reduced to rusty junk.
So, given randomly precocious development in the more ancient civilizations, the whole Classical era of the Old World could be massively transformed (butterflying away all the better-known societies and languages of course). I'd think a major change would be the nature of war at sea--until the English pioneered the use of naval cannon to pound enemy ships apart at fairly long range, naval combat was essentially a matter of ships physically coming together and their crews fighting it out on the decks.
Ancient ships of course were much less well suited to being platforms for heavy guns than the sorts of vessels Early Modern Europeans could make, and necessity is mother to invention only to an extent; I don't think people like the ancient Phoenicians could have made a proper galleon, let alone a ship of the line, no matter how much the idea of seagoing castles might have appealed to them.
Another major thing to consider is that long before either suitable powder for guns nor suitable materials for reusable gun barrels make their appearance, rocketry might substitute. Rockets can use forms of gunpowder that aren't useful for guns, and it is possible that before guns are developed the arts of rocketry might become elaborate and widespread, possibly developing "candy rockets" and the like. (This diversity of rapidly combusting/decomposing substances might also open the way to forms of "gun" "powder" that would develop only late OTL).
I've wondered whether sufficiently advanced solid rocket fuels might be developed to deliver a low but steady and long-sustained thrust, that could lift gliders up to significant heights, allowing human flight (on various scales but limited range) thousands of years ago. In warfare, such gliders would probably be mainly useful as scouts, rising up to fairly high altitudes for long lines of sight, perhaps venturing over the edge of enemy lines, then gliding back to the safety of their own lines with intelligence. At sea such scouts might not only serve in combat but also just to assist navigation by sighting distant land to steer for before supplies run too low.
Possibly, given thousands of years of lead time, practical forms of heavy rocket artillery might be developed, to drop seriously sized bombs more or less on target over significant ranges--probably not beyond line of sight, and certainly aiming is a problem unless the rockets have human riders to guide them (not necessarily kamikazes--they might jump off with parachutes or gliders to either try to fly back to their side or come down as shock-paratroops--the latter option does strike me as tantamount to suicide if their foes have numbers and are not daunted, while the former would also be very risky--definitely a berserker type of warrior if not outright suicidal!
)
Aside from speculation about the ramifications of really ancient development of gunpowder and/or rocketry, we might instead consider that the improbability suggests that the Chinese do
not stumble on it--at all, or the practice dies out in obscurity and either someone else does (early, so the practice is radiating out of a different center--West Africa say, or Europe--it could easily be India but then the long-term outcome might be pretty similar to OTL overall).
Or consider an Old World that never does stumble on it anywhere, at least not until some developing capitalist (or some other conceivable center of sustained innovation) society is systematically developing chemical science. No guns and no rockets and no pre-modern practice with explosives of any kind would imply significantly different conditions in all of developed Eurasia and Africa; by the time the early modern European explorers reached the Indian Ocean, both they and the peoples they troubled there had been adapting to gunnery, artillery and rockets for centuries. As I mentioned, naval warfare was transformed; that would not be the case here (even if such methods as "Greek fire" became widespread they would imply a different sort of sea battle than artillery does). I suppose all the major civilization centers of the Old World would remain strong and developed, but there would be some shifting of power. For instance, the Asians of the steppes might remain stronger relative to the civilizations, and the overland Silk Road might tend to dominate more over seaborne trade insofar as they can maintain consolidated empires spanning the length of the trade route. I would expect Europe to still be on a path toward something close to Early Modern capitalism, but the details of statebuilding might be quite different due to the lack of artillery or the prospect of arming infantry with any sort of gun. (As the general technical level of development rises I suppose crossbows and such will become more common). It seems likely that a more medieval sort of warfare would prevail longer, though my imagination may be falling short of ingenious evolutions that people would think of in the absence of gunfire.
It might well lead to a delay in the eruption of Europeans onto the global scale, due to the cumulative effects of all these factors above--that is, delayed centralization of strong European states (perhaps); lack of transformation of naval warfare conceivably hobbling the rate of development of navigation in general; persistence of Central Asian cavalry empires diverting trade to continental overland routes, which favored Europe versus the OTL shift to seaborne trade that put the major routes where the Ottomans (themselves an example of a "gunpowder empire," here transformed or butterflied away completely) might deny the trade to Europe, thereby motivating the development of sea trade.
If the outpouring of Europeans overseas, into the Western hemisphere and horning in in the Indian Ocean and Indonesia, is delayed and slowed enough, perhaps capitalist civilization as we know it might be preempted completely due to European political powers gaining the upper hand over their mercantile classes.
I actually don't think that's too likely, and that the Europeans will come out only somewhat delayed if at all, perhaps from an even more fractious and divided Europe, but still find the Americas and insinuate themselves everywhere.
But now what if our gunpowerless Old World Europeans find an America where the major civilizations
do have gunpowder,having stumbled on it a couple thousand years before?
The Americans still wouldn't have draft animals to speak of, and would I would think still not be more advanced in metallurgy than OTL, so probably they either would not have guns at all, or else have those rare Bronze Age ones only great warlords can afford. I think the major use of anciently developed gunpowder forms in the Western Hemisphere native societies would rather take the forms of rocketry. Perhaps even including rocket-lifted glider flight, so that corps of flying warrior/craftsmen would be ancient and widespread.
Now I don't think having gunpowder alone would reverse the relationship between the Old and New Worlds; the invaders from Eurasia would still bring a potent brew of diseases with them after all, and even with gunpowder weapons I don't think the Native political structures would be easier to centralize, with no draft animals. A major factor that might slow down and buffer the decimation the Columbian exchange brought them OTL would be if the Europeans are indeed less politically centralized, if instead of expeditions backed by the deep pockets of Ferdinand and Isabella (or rather their Genoese and German financiers) they are a horde of relatively limited adventurers coming in random dribs and drabs. Presumably what strong realms do exist in Europe (England seems likely to be one) will attempt major projects of conquest and consolidation, but in competition with many smaller powers. This might give clever Native politicians opportunities to negotiate better terms.
Meanwhile, in addition to perhaps a couple of diseases, tobacco, chocolate, and a whole lot of American food plants, now the eastward branch of the Columbian Exchange is introducing gunpowder to the Old World. If the Native Americans have not actually developed guns as such but rather rocketry, it might take a while for someone in possession of Old World metallurgy to think of trying to make any, and the state of the art would of course be centuries behind OTL. It would also be flowing from the European Atlantic coast east and south. Conceivably it might not be Europeans but one or more of their rivals, such as the various Islamic powers or the West Africans, who first try it and perhaps turn the tables on the Europeans?