Let's say for the sake of argument that the Western Roman Empire was able to not only survive but last in some capacity until the modern day (at least inside Italy) and the Eastern Empire falls sometime around the same time as the WRE in OTL. What impact might this have on European history and World history as a whole? Due to the WRE surviving, might we see a Roman Colonial Empire in the New World and elsewhere? How might having a WRE that never totally falls affect the Catholic and Orthodox churches?
Gonna break this down so I can better hypothesize. I'm also gonna assume for the sake of discussion that Western Rome held the equal amount of territory to OTL Byzantium:
- A surviving Western Roman Empire would in all likelihood make an attempt to reclaim the eastern lands, much as Justinian II in OTL did with the western lands. Its possible that a Justinian-like emperor would do even better in the east than Justinian did in the west, although there is always the chance that Egypt could assert its independence under a dynasty of mixed Greek-Egyptian-Roman heritage-possibly expanding into the Levant and south to Nubia. Should this happen, West Rome would likely try and defeat them, possibly conquering the Levant and imposing vassalage on them. The West Romans could make the Frankish state a foederati and use them to bring Britannia back under Roman rule. Hispania could also be reconquered for Rome. There would likely be no Dark Ages as this was predicated on the collapse of Roman rule across Western Europe. They would still have to contend with Vikings descending from Scandinavia and while military fortunes would be mixed in the Roman-Viking Wars, there would be insufficient manpower for either side to deliver a knockout blow as Scandinavia is sparsely inhabited and Rome would likely have to divert troops east and south to try and stem the Arab advance. If the Avars, Slavs, and Magyars irrupt into Europe as in OTL, Rome could lose some border territories around Pannonia and Transcarpathia to them but I suspect that they would make foederatii of the invaders as well. Egypt might lose the Levant initially to the Arabs, but should Rome send sufficient forces into the region, the Levant could be liberated. There is some possibility for a joint Egyptian-Roman expedition into Arabia, and should this be successful, Islam would become a minority sect as opposed to a world religion as in OTL. (The second consequence of this successful expedition would be that Persia would likely regain any territories lost to the Arabs and become a major power in the Middle East again).
-Assuming that Western Rome eventually incorporate the various foederatii (Franks, Alemanni, Saxons, Burgundians) into the empire, they would rule Germany up to the Elbe River. This could play a role in the later development of states such as Pomerania, Poland, Great Moravia and Hungary, either resulting in them becoming allied to Rome or client-states. Regardless, they would have protection against the Kievan Rus (with them even possibly introducing Roman Christianity to them), Fast-forward to the 1230s-1240s and the Mongol Invasion and these states, allied or aligned with Rome would be better able to contain the Mongol attacks, though with substantial losses of life (Kiev would likely still be razed to the ground by the Mongols). The Black Death would still spread into Europe as a result of contact with the Mongols as in OTL, and likely the same total in dead would result (hard to determine in an alternate scenario).This could, however, begin a trend of separatism in the various regions as each region could be affected by the Plague in different ways. That separatism could even be given a boost with the divergence in religious views which would start off quietly and in affected villages, but gradually spread across the provinces.
-A Roman colonial empire would be possible assuming that they maintained their rule in Hispania. Where they would first establish settler colonies would depend on conditions such as ocean currents and weather. Assuming Brazil is discovered first (even before the OTL voyages around Africa), this would lead to an earlier discovery of the Americas. Because Western Europe (except Scandinavia) would be united under Roman hegemony, the only competitors for the New World would be the Scandinavian peoples-assuming the Chinese and/or Japanese haven't come upon the Americas around the same time due to various other butterflies which are too many to list.
-As far as a Protestant Reformation, none as we know it in OTL would happen. But should slight divergences in religious practices happen as a result of the combination of Plague, contact with native peoples in the Americas, or the lingering traces of Islam making its way via Egypt into Rome, then a reformation would be likely. I don't honestly see an Orthodox religion being a factor as the eastern lands had not been allowed to remain sovereign long enough for the Great Schism to happen. But then again, Kievan Rus (if it somehow survived the Mongol Invasions and even incorporated Tengrist shamanism into its Christian practices) could forge an orthodox version of Christianity.
-As for how long this resurrected Roman (Western) Empire could last, that's harder to determine. Even with all the hypothesized possibilities I listed, there is no certain path forward beyond the Renaissance. Separatism could become such an issue in that time that provinces break away and become national states. Client-states could see rebellions which install a more independent government that could even go to war with Rome. Generals such as Marlborough, Conde, and even Napoleon could still become generals or they could have different career outcomes. Assuming that Rome loses all but the original core territory (Italia) in the likely chaos, the Romans might be willing to align with any power that promises their restoration to greatness a la Mussolini.
There are likely others who can better outline possibilities than I just did, but this is at least my attempt to theorize based on what I know, and while I did not go into exacting detail, the general idea should hopefully be sufficient enough.