Since no one else will reply, I will give my opinion even though this is far from my area of expertise.
As far as the Saharan pump theory, if a civilization was to form there, it would be situated along the rivers connecting to the Nile up to the Libyan coast forming a network across the Eastern Sahara of fishermen and farmers along the shores of rivers, likely too with villages on the exterior of these rivers close to the likely Savannah surrounding. Cities would still be along the river but would look essential like an extended Egyptian river going culture but with more variety into the the Savannah. If possible and depending on the size of the river systems, this could connect Egypt to West African gold as well as that found in Punt and Kush.
However, this situation of prosperity is inevitably short and the mass exodus and human suffering caused from the rapid desertification is frightening, likely this would cause a dark age as refugees from the disaster would wreak havoc upon cultures in the way of its migration, with the assumption that large populations are created due to large scale agriculture.
Likely in this situation, we would see land covering the Persian gulf, with the extended Tigris and Euphrates passing Basra and reaching down to Dilmun and perhaps Suhar. Thus the myths of the Sumerians would be a reality of sorts, in how a powerful civilization exists in the fertile land to the south of Eridu and contemporary with the famed land of Dilmun. What is here is anyone's guess, perhaps in this pot of land is the origin of the Sumerian tongue and maybe the Elamite and Kassite isolates. In this area we see a much larger Mesopotamian culture group extending far to the south and even more fertile than later. This civilization would likely surpass the one in Africa in terms of prosperity as the distance to India would be shorter.
However, once again, the ice caps will melt and recede in cycle and the result will be the wholesale flooding of these lands, destroying these civilizations. These refugees due to the nature of the disaster, likely do not destroy in their path as they are carried into the ocean or submerged.
India would definitely have more coast line, meaning cities outward. Also the distance from the early Indus Harappan will be shorter to the people of Mesopotamia and the cultures therein, thus by virtue of push and pulls, will create more trade societies to take advantage of the shorter distance.
Then of course there is whatever is going on in Göbekle Tepe, the societies here would become advanced in response to the civilization to its south (assuming they aren't just as advanced otl, which we will likely never know).
Frankly we do not quite understand civilization before Sumeria, the possibility of advanced societies (without writing), in the manner of the Inca is entirely possible, as their remains would be submerged in water, or eroded away or possibly used as material by later civilizations. In all honesty it is easier to discover bones of extinct animals than it is to decipher the history of human civilization before Sumer.