Been doing some more research on this and it's helped me to refine my ideas on this.
Basic idea is the same, in that Andronikos III avoids his untimely death from malaria in favor of a later one in 1347 from the first wave of Black Death. Under his extended reign of a further six years, the Latin Principalities are vassalized (uniting Greece under the Romans for the first time since 1204) and the expansion of the Roman Navy continues; apparently both Alexios Apokaukos and John Kantakouzenos were in agreement on this despite their growing feud at the time and John attempted it after his victory. Instead of 40 ships in 1341, it's entirely possible to see 60 or so by 1347 I would think, given the improving economy and united Imperial focus. Without the Civil War, both the Serbians and Ottoman Turks will be kept at bay for years longer; the Serbs at the Battle of Stephaniana were defeated by just 3,100 Turkish while Kantakouzenos was able to raise a force of 6,000 men in 1341/1342 historically. Given that, and the Roman Navy being able to keep the Turks at bay in Anatolia, is why I make this conclusion.
When the Black Death comes and Andronikos finally dies, John V will be either 15 or 16 and thus able to claim the throne in his own right. Given the influence of his mother and the Patriarch, who historically were part of the reformer faction that was the Regency, it's likely he would favor Apokaukos over Kantakouzenos (And is my chief interest in this PoD). I don't know if this will still result in a Civil War; the Emperor ruling in his own right removes the chief excuse of Kantakouzenos and the nobles to revolt, although that alone doesn't stop them from trying on its own. Maybe John gets arrested or assassinated, removing the natural leader and we instead get a serious of disconnected nobel revolts that gets rapidly put down by the new Emperor's loyalists. If it does still result in a civil war, it will be short; the Regency IOTL had the Church, the commoners, Constantinople and the Navy because of Apokaukos. John had the Army and the Nobility, but the Emperor being in his majority would weaken John's historic influence on the Army.
If war still comes, I suspect it will end quickly. The more powerful Roman Navy means the Aydınids are removed from the board and Thessaloniki can be supplied and kept under Imperial direct authority much easier than OTL. With the Emperor in control of Constantinople, Kantakouzenos will still need to withdraw to Macedonia, both to assemble his forces and put down peasant revolts spurred on by the Emperor/Apoukakos. He will be unable to advance on Constantinople until the threat of Thessaloniki in his rear is dealt with, which seems unlikely given the Imperial Navy can keep it supplied, thus buying the Emperor time to raise his own army of loyalists. As soon as the Emperor marches, it seems likely the result will be decisive, given the Nobles will be caught between the anvil that is Thessaloniki and the hammer of the Emperor's Army. If they retreat, that means surrendering much of their territory to the Emperor anyway, and thus likely collapsing the morale of their forces. In short, if a Civil War does happen, I suspect it'll be over between 1347-1349, thus fairly quick and much less destructive with no Turks in play in Europe either. If the Serbs do try to meddle here, the collapse of the Kantakouzenos side will enable the Emperor to drive them off quickly. From here, things get more interesting.
Historically the Byzantines and Genoa fought a war in 1348-1349. It was indecisive at the time, but the chief Byzantine goal of revoking Genoa's trade concessions was not achieved. This war was only fought because the Empire was desperate for funds, and with a less destructive civil war, they have the ability to avoid it. This is chiefly because the power of the nobility would've been broken by the civil war/removal of Kantakouzenos, enabling the Imperial authorities to revoke many of their privileges and thus increase the taxation upon them. Any who do revolt can have their lands confiscated for their treason too. This will help to make the Empire more financial solvent, but the desire to revoke the Italian trade concessions is still a goal and was a main reason for the naval expansion advocated by both sides. Thus, I think the war will still come, but on Imperial terms...and that means Byzantium will join with Venice under more favorable strategic conditions in the 1350-1355 war.
Venice would be delighted to have the 60 warships of the Empire on her side and to seek to cut Genoa down a peg, which means the Byzantines will have the opening to not only take Galata and revoke the trade concessions, but also to supplant Genoa in the Black Sea and Aegean. Thus, the Empire can reclaim islands like Lesbos and Kaffa/Feodosia in the Crimea. When the dust settles, not only will the Empire finally have control over trading policy in its own territories, but it will also be able to start undertaking its own trading zone as I talked about in my original post. If they play their cards right later on, they can also pick off the Venetian possessions in the Aegean, including possibly Crete if they get really lucky, in the Third Genonese-Venetian War later on. At that point, the Italian winner of that conflict will have to start looking at the Byzantines funny, but the Empire will likely be sufficiently strengthened and their new enemy weakened, that its hostility is survivable.
In short, we now have an Empire controlling everything under the Haemus, with its finances well off via the reduction of noble power and the reclaiming of trading policy from the Italians. A nascent trading bloc will also be forming under their auspices in both the Aegean and Black Sea, which will further their financial recovery and thus enable the Empire to bolster both its influence and military capabilities. It won't be as strong as it was before 1204, obviously, but a regional great power still seems possible. It'll probably follow a policy of close ties with Trebizond and Georgia to balance the Turks in Anatolia, as well as the traditional "Divide and Conquer" strategy, to prevent any one Turkish polity from becoming too powerful as to dominate Anatolia. I can see them allying with the various Emirates to cut down the Ottomans at some point, while also seeking to use both soft and hard power to protect the remaining Greek and Armenian Christians as much as possible in Asia Minor. From here, a lot could happen.
I suspect the PoD will remove Timur, simply because of how close he was born to the PoD. If he is still coming, than a Byzantine reconquest of Anatolia actually becomes possible given how much Timur is likely to cripple them and the Roman Navy means he can't get at them either, enabling them to take advantage of the post Timur chaos. As I said, I doubt he will come which leaves a lot open. Earlier I speculated the replacement of the Ottomans with the Orthodox Byzantines would cripple the Tartars and enable a faster Russian conquest of OTL Ukraine and the Kuban. It's actually an even better situation than I thought; without Timur's help, there's no rise of Tokhtamysh and subsquent re-unification of the Blue and White Hordes into the Golden Horde. Now, Timur did smash them anyway historically, but here they remain unstable and disunited and their local Christian neighbors were already gaining the military edge with things like early artillery. Timur's swathe of destruction is what also pushed a lot of the new Tartars into the Crimean Khanate OTL, as he destroyed their power base along the Volga. Disunity and Russian military advantage will still see this happen, but without an Ottoman patron and a more gradual collapse as well as the ability of the Christian states in the region to block this Tartar "retreat" into the Ukraine, the Crimean Khanate as we know it will never happen.
I can thus see the Steppe and up to the Caucasus under the control of the Russians (Or PLC, in the case of much of Ukraine) by 1600. Byzantine trading influence and naval support will assist in this, both because Constantinople would like to be able to project influence in the region via their fellow Orthodox Russians as well as the trading opportunities inherent. An earlier conquest of the Steppe and Byzantine support will also strengthen the Georgians too, thus also increasing the Christian hand in the Caucasus; I don't think it's unlikely to assume the Russians here will convert to Christianity most of the Steppe people and, in tandem with the Georgians, do much the same in the Caucasus. I think the Azeris would end up the long holdouts, due to the Persians and local terrain.
Final prospects I see for the Byzantines and their Orthodox pals comes in varied places.
The Mamelukes are still going to be on the decline and here there is no Ottomans to pull them down, but there are the Byzantines who are a waxing power by the 1500s. A repetition of Manuel's efforts in the 1160s could happen given the obvious economic and prestige benefits, while the Copts are also still 30-40% of the population most likely at this time, thus giving a strong support base to leverage. A Byzantine reconquest of Egypt would in the long run strengthen the Empire and also the Copts; when Egypt inevitably goes independent at some point, it will likely be majority Coptic Christian given centuries of Imperial favor to them and the removal of Muslim mechanisms of conversion.
Finally, it's likely the Shia sects that historically converted Iran are removed from existence here and the Christian communities of Iran, which probably constituted 25-30% of the population before Timur slaughtered them, are still extant. One idea I've had is that if a Time of Troubles analogue still comes for the Russians, a confederation of Cossack tribes band together and move South as TTL's replacement of Timur, conquering the weak Sunni dynasty of Iran and replacing it with one of their own. They'd probably assimilate into the native Christians, while advancing them as a "native" Iranian faith given Nestorianism pre-existed Islam and was on the ascendancy before the Muslim invasions cut it short. It's possible Nestorianism will become syncretic given the Cossacks are Orthodox, meaning the Nestorian Iran will be able to become closer to the wider Orthodox world.
If there isn't a Time of Troubles like event, and with the Steppe in Europe mostly secured and being settled, the Tsar may turn the Cossacks into Central Asia in the 1600s, which would likely result in that region becoming majority Orthodox in the long run.
Very interesting! Re: Russia: By the 21st century, just how far south will Russia extend/expand?
