If one is interested in an ancient POD, lately Jonathan Edelstein seems to be hot on the trail of bases for high civilization in pre-Classical times in West Africa--considering whether rice, or other crucial cultivars, might be developed earlier there, and whether the OTL Nok iron-working cultural group might have begun earlier or been more extensive. I suppose soon he might combine several of these to have a West African constellation of imperial and related states antedating the rise of Classical Greece, or even the better part of a thousand years earlier. Presumably the ancient states will rise and collapse, but there will be successors.
If this were the case then it seems likely to me there would be interactions, if perhaps weak and indirect, with Classical civilization in the Med (and Egypt, and perhaps as far afield as India). How this relates to the OP depends on one's view of butterflies; by the orthodoxy espoused by many if your POD is in 3000 or 4000 BCE, then by not many centuries after that every society in the world must develop along increasingly divergent ATL lines, since "different sperm hits different eggs" thanks to the pure chaos in changed weather caused by different human choices, not to mention actual chains of social interaction. I have a dissenting opinion, that mere chaos can always be factored out if one so desires since alternate timelines do not necessarily arise from a line that would but for the POD have led to OTL--that at any time in the past one likes there are already a practical infinity of timelines, and the POD could be in another of them that would, but for the POD, have diverged from ours due to chaos--but it so happens, we pick the sheaf of timelines that have evolved correspondences to ours at some later time of interest, highlighting the logical effects of the divergence and canceling out, as much as is logically possible and to the degree we like, merely random divergences from our own.
Thus, a West African ancient civilization will cause random changes in weather, but we can suppose that but for these changes, the randomness of that ATL would have diverged from ours far away from West Africa--but here the butterflies work to cause convergence instead of divergence. Also an agricultural civilization existing where one did not OTL might cause systematic climatic shifts--attempting to irrigate accessible parts of the Sahel might change the persistent moisture balance for instance, and cause persistently different weather perhaps far from their own site. We may or may not be able to compensate for that too. Finally the ATL society will interact, as a society, with its immediate neighbors who will be affected and changed, and this will transmit a wave of changed culture, society, economics and politics. But we might also set reasonable boundaries or ranges beyond which these changes amount to merely random perturbations, and so factor them out too beyond that range.
Thus, we might reasonably imagine that despite this extra sheaf of societies and polities evolving alongside the OTL known Classical world, if it is far enough away the canon peoples of OTL still develop much as OTL, right down if we like to having the same kings and merchants and philosophers and so on--the Sahara and the indirect path via the southeast Sahel boundary to Kush and Egypt are distance enough to leave the Mediterranean world pretty much as it was.
At some point this "antibutterfly" choice of timeline must break down of course; at some time the peoples who were once so distant from ATL West Africa are no longer out of range. Thus presumably Mediterranean explorers such as the Phoenicians might venture coastwise along the African coast and visit the ATL cities and nations. They might have little impact because traveling both ways by sea is difficult; the classical story of the circumnavigation of Africa by Phoenicians commissioned by an Egyptian Pharaoh is that because coming back north from West Africa is not so easily done; the expedition found it easier to keep on pushing south until they rounded the Cape of Good Hope, and then found themselves going north again and eventually back to familiar Red Sea waters. Until someone, possibly if not indeed probably the West Africans themselves, develops the various arts of seamanship and shipbuilding that permitted Europeans OTL to survive a deep ocean crossing and have confidence they could find a way home again, the Mediterranean and West African spheres will find contact by sea chancy and difficult.
There is the overland route over the Sahara, but that is controlled by middlemen, desert dwellers who live in part off of the trade across the desert. With more people living in a more developed set of societies south of the Sahara, presumably trade along those routes will be increased and this will somewhat raise the fortunes of those desert peoples; it may also attract the attention of strong states north or south, such as the Romans if we don't butterfly them away. Or, if the basis of high civilization in West Africa is strong enough, maybe an African high king or republic or something of the sort intrudes via conquering or co-opting the desert folk into Mediterranean politics and history.
I've followed a timeline which supposed the Romans could have been motivated to conquer West Africa and capable of doing so, by overland movement mainly. I think here the motive for contact would be greater than OTL, but the strength of West African peoples in their own homelands would deter any Roman projects of conquest, or stop them if attempted--the Romans might get lucky and stumble in during a warring states interregnum and find local allies enough to establish themselves as imperial overlords, perhaps. Or they might merely open the door for a strong West African political entity to intrude on the disintegration of Roman power in North Africa west of Libya.
Given the distances, the desert, the insular interests of the desert peoples, we might call it a push and simply settle for the two regions having some knowledge of each other and limited trade. If we assume the Christian religion is not butterflied away, I suppose missionaries would head there and probably make some kind of progress--perhaps the West African lands become anciently Christian, perhaps like Persia they largely decline to do so, perhaps very interesting synergies of Christian and West African belief systems emerge.
Given all this, we could well have Western European civilizations develop much as OTL even though these West African lands and peoples are rather well known to medieval scholarship and indeed to visitors who voyage there by land or perhaps sea, all the way to the point where the Portuguese start perfecting their navigation with the ambition of sailing around Africa.
At that point, I'd say that we have to cast aside what remains of the butterfly net and consider that beyond it, European civilization must develop differently. The Portuguese, and other Europeans who come down the African coast after them, probably cannot act just as they did OTL. To be sure by the fifteenth century of OTL there were some remarkable civilizations there and they were nevertheless drawn, largely by European hunger for slaves to use in the plantations of the Caribbean and South America, into a comprador relationship where warlords gained strength with European firearms in return for becoming reliable slave sellers; depending on the general level of technology in the region this might still happen. But it also might not, if there are strong states and a somewhat higher level of technology; they would not be dependent on Europeans for as much and might be more advanced in some respects, very likely having their own well-developed gunpowder weapons and a high level of metallurgy in particular. A local potentate or even a set of rival states might insist on more favorable terms of trade.
Having contact with Europe right as it is on the cusp of modernity, and being itself a peripheral region to the centers of world civilization just as Europe is, the region might well be drawn, either by semi-integrating with the expanding European system or (if for instance the region had long ago converted to Islam, or holds staunchly to a third religious system that the Christian Europeans consider hostile) in rivalry. They might pre-empt the oceanic trade routes themselves, or anyway participate in them on their own hook.
Indeed, an ancient set of civilizations in that region seems rather likely to me to at some point develop their seafaring abilities on the Atlantic rather earlier than the Europeans do, and being quite close to the easternmost extension of the Americas, to have stumbled at some time long before 1500 on the western continents and to be in continual contact with them. I can see the possibility, if they can learn how to travel to eastern Brazil and return reliably to West Africa, that they will explore and trade up the Amazon and thus gain crops and techniques to develop the Congo using Amazonian methods. The disease exchange will probably still be devastating in South America but perhaps more drawn out over time, allowing some Terra Prieta civilizations in the central continent to survive and interact.
Thus there might not merely be a West African potential for co-evolving with a greatly diverted capitalist system, but a whole equatorial Atlantic system to interact. The odds seem pretty good to me that somewhere in there, in old West Africa itself, or in the Congo, or Amazonia, or a transformed Caribbean, for nations fully equal with the most advanced European nations to arise and participate, as members of an expanded European system or as rivals of it, in the rise of industrial civilization.
I suppose this is not quite what the OP gets at, since it assumes that modern Wesphalian industrialized nations arise first in Europe, and some overseas region belatedly catches up to it Meiji style. West Africa is far enough from Europe and its ancestral civilizations that I can imagine the latter arising as per OTL up to 1500, but as it is the first step taken on the road to world trade and dominion by the Portuguese OTL, it has to strongly influence what happens from that point on, and if the early centuries of the interaction do work out to correspond to OTL I'd say the potential for a radical difference would be preempted. Odds are, given how disadvantageous the terms the Europeans eventually imposed on the region were, that things will necessarily go differently from early on, which involves the West Africans in the rising European system at the very beginning, as central participants. They might, instead of industrializing themselves, prevent the Europeans collectively from having that potential, by blocking their routes eastward and preempting their colonies westward.
But I'm rolling here with the idea they wind up being an integral part of the Atlantic meta-society and involved in everything the Europeans did OTL, taking their place as one (or more) of the capitalist powers, perhaps changing many details of just how the general process of incorporating the world in a global capitalist system happens.
It would be interesting if they refuse to allow the trans-Atlantic slave trade to arise, anyway on European terms--they might do something like that themselves in the Americas, putting their own criminals or other Africans captured in warfare to work on plantations they own in colonies they operate, and yet refuse to sell such workers to European would-be buyers. They might even take it upon themselves to pre-empt or dislodge European strongholds on the coast of Africa, moving down south past the Congo to secure Angola and Namibia, or anyway their coasts, depriving Europeans of all practical options of getting African labor except on voluntary terms.
If the European powers want the wealth that went OTL to the owners of the plantations, they'd have to use European labor to get it--as they did for a while OTL. Indentured and more strongly bound convict labor from the slums of London or Paris would be more troublesome than African slaves, due to their ability to act within the established social and political systems; the danger of slave rebellion is higher and to have them working at tropical plantations is to remove them from their OTL labors in Europe or settler colonies; even if the European capitalist system is not aborted completely its growth must be slower--on the other hand the West African-connected peoples will bring more total population into the mix. They too might grow more slowly per capita because they won't have the advantage of "alien" slaves they can work to death without internal repercussions either, but the total base from which it grows more slowly is larger.
I wonder if it could happen that while Africans sell their undesirables or third party captives to Europeans to work their plantations, the Europeans also sell their criminals, "surplus," or captives in wars from weak European nations to Africans to work them on their plantations, and the Atlantic economy rises on the basis of slavery as per OTL--but with whites enslaved just as blacks are, and the whole system emerging as one where polarization is seen directly and simply as a matter of wealth and social position, and race is never seen as a strongly relevant factor.