With the Early Dynastic period it's a bit difficult to assert exactly what was going on...
The order imposed by the First Dynasty appears to have broken down under King Anedjib, who was in turn succeeded by Semerkhet under dubious circumstances (it has been hypothesized that Semerkhet was a usurper, and he took vindictive measures against Anedjib such as having his name erased from votive vases). Later dynasties appear to have regarded Semerkhet's succession and 9 year reign as legitimate, however, as his name is included in all king lists. Semerkhet apparently suffered several "disasters" (of an unspecified nature) during his reign, was then succeeded by Qa'a, who actually ruled for 26 years (give or take) despite apparent unrest in both the South and the North - implying a multi-faction rebellion against central authority rather than a war between two different cults.
The Second Dynasty begins with Hotepsekhemwy, whose name ("Both Powers are Content") implies that he had a hand in stabilizing the country. He also buried Qa'a, which implies that he regarded him as the legitimate monarch on whose authority he based his own claim. The dynasty is actually stable (and seems to have succeeded the First in good order) until the reign of Set-Peribsen, whom Jonathan Edelstein mentioned. There are theories that Set-Peribsen was a heretic attempting to completely re-work Egyptian religion (which was still in its formative period), to the point where his serekh-name was shown under the protection of the Set-animal rather than the falcon of Horus, but the fact that his mortuary cult persisted until the Fourth Dynasty suggests that, whatever his faults, heresy was not one of them. Some scholars think Egypt was actually divided at the time (either at the beginning of or shortly before Peribsen's reign), with Set-Peribsen actually ruling in Upper Egypt from Tjenu (Thinis), and another king, Senedj, ruling in Lower Egypt from Mennufer (Memphis).
If this latter hypothesis is correct, it may actually be the source of the Horus and Set myth (in which Set ruled Upper Egypt and Horus Lower Egypt, despite their traditional associations being reversed). It's possible that Senedj and Set-Peribsen divided Egypt between them in the wake of civil unrest following the death of Nynetjer, and even seems likely that Set-Peribsen's successor, Sekhemib-Perenma'at, only ruled Upper Egypt as well... The country was not firmly united under a single ruler until Khasekhemwy ("Both Powers Arise" - another possible reference to Reunification - especially as Khasekhemwy was originally named Khasekhem, "Power Arises"), after which a unified Egypt is firmly established by the Third Dynasty - the founding dynasty of the Old Kingdom.
TL;DR - There may have been a war between kings associated with Horus and Set, but their power bases were reversed - with the Horus faction in Lower Egypt and the Set-faction in Upper Egypt. If this was the case, it actually was Lower Egypt that reunited the country under Khasekhemwy.
Regarding a triumphant Cult of Set in this war... I agree and disagree with some of what Jonathan said...
I don't think the centre of political power would shift to Lower Egypt - Mennufer was already surprisingly well established as the capital of a united Egypt by this time, so the victorious faction would want to legitimize itself by moving the capital back there.
I don't think human sacrifice would be practiced any longer than OTL - it had already ceased for three generations by the time Set-Peribsen and Senedj took their respective thrones
I think relations with the Asiatics would actually be much the same. It was the Horus-kings of Lower Egypt that began the "special relationship" with Byblos, not the Set-kings of Upper Egypt. In fact, Khasekhemwy was the king responsible for essentially founding the institution of a navy in Egypt, based on imports of cedar from *Lebanon. A victory for the Set-kings in this scenario may actually sour Egyptian-Levantine relations for a while, and Set's status as patron-god of foreign lands may actually be re-worked to make him the vanquisher of the foreigners who aided the Horus-kings.
I do agree that the Osiris myth would be reworked to paint Set in a more flattering light. It's also possible that we might see an order of kingly succession based more on seniority rather than primogeniture, but that's operating on the assumption that Senedj was Nynetjer's son, and Peribsen Nynetjer's brother (which would have it fit nicely into the Horus myth), their conflict ultimately stemming from a succession dispute.
As for what Fearless Leader said in the other thread...
- Hieroglyphs aren't going anywhere. They predate the Early Dynastic. Also, the Phoenician script is ultimately derived from the Egyptian hieroglyphic script (via proto-Sinaitic and Canaanite), so getting rid of hieroglyphs in the first place means it never emerges at all.
- The Pyramids as we know them certainly wouldn't appear, though chances are something pyramid-like would. There are only so many ways to build a tall structure with the technology available in the period.
Now, broad historical generalizations? Maybe in keeping with Set's changed role as vanquisher of foreign lands (assuming the hypothesis about the war is correct), Egypt becomes more interventionist in the Near East earlier in its history. This could go either way, but regardless, history as we know it is completely butterflied. I think Jonathan may be on to something about hypothetical future Levantine dynasties of Egypt syncretizing Set with their own storm gods to make their rule more palatable, but that's far into the future here...