Remember Christianity was founded in the east, in the Roman province of Judea. On the eve of the Arab invasions, the Roman Empire, which by the 7th century was mostly the eastern empire with a few holdings in Italy and Spain, sponsored what would become the Catholic church. The Germanic rulers of western Europe on the other hand predominantly followed the Arian heresy.
Next consider the Muslim invasions, events as transformative as they were unlikely. Arguably the only reason tribesmen from the backwater of Arabia managed to conquer most of the western world was the epic war which the two main powers of the time, the Roman (or "Byzantine") Empire and Sassanid Persia, fought from 602 through 628. That war thoroughly exhausted both sides in terms of money, infrastructure, and personnel, allowing the Arabs to sweep through both empires just a few years later.
Removing that Byzantine-Sassanid War is the key to meddling with the geographic placement of Christianity and Islam. So how can that be done? First, we must examine why the war happened in the first place. Up until 602, the emperors of Rome and Persia, Maurice and Khosrau II respectively, were on very good terms. Maurice had rescued Khrosau II from a rebellion a decade earlier, for which Khosrau owed him gratitude. So when Maurice was assassinated by his troops in the winter of 602 for ordering them to winter north of the Danube River, which meant exposing them to attack by invading Slavs and Avars, Khosrau did not hesitate to attack.
Let's say Maurice chose not to push his troops and allowed them to winter in friendly territory. Then for the next three decades, a state of relative peace is maintained between Rome and Persia. Maurice is succeeded by his son and co-emperor Theodosius and things are fine and dandy until the 630s, when, as in our timeline, the Muslims invade.
The Muslim armies were led by a strategical genius named Khalid ibn Al-Whalid. Al-Whalid, reading the different situation, likely would not have attempted to invade Rome and Persia simultaneously. I could come up with any number of justifications for invading Rome first, but to save time let us just assume he decides to head west first. Arab armies, not overstretched as one might argue they were in our timeline, still manage to defeat Roman forces which are more lively then in our timeline. The Levant and Syria fall in 637 and Egypt falls in 639.
But rather then rapidly expanding from this position, war with Rome and the effort of pacifying the newly conquered peoples is enough to keep the Muslims bottled up in this area. Attempts to invade Persia are beaten off by Khosrau and then his successor. But perhaps a decade later, the mass of Arab tribesmen, much more concentrated here than in our timeline, conquer North Africa. A few decades later they cross the Straights of Gibralter, conquering the Visigothic Kingdom in Spain, then storming across the Pyrenees to beat the Franks into submission before turning south to sack Rome itself.
All this might seem far fetched, but remember the resources which were expended in Persia are free in this timeline, those extra Arab tribesmen itching for their own land to rule. But I imagine sacking Rome would have been a step too far for our ambitious Arabs. Such a move would likely have sparked a revolt in their eastern territories of Syria, the Levant, and Egypt. With all of their armies in Western Europe, such a revolt would have been successful. Rather than returning to the fold of Constantinople the Copts of Egypt would likely have created their own kingdom. They had long been ostracized by Constantinople for following Miaphyisitism, a Christian sect deemed a heresy nearly 200 years prior to the Arab invasions.
With refugees from the east to bolster their armies in Western Europe, the Muslims certainly would have been able to hang onto Spain and Gaul along with North Africa. A full conquest of Italy also seems likely, as a Byzantium without Egypt would be in total crisis mode, struggling to feed Constantinople, much less regain loss territories to the West. So there you have it, Western Europe and western North Africa is eventually Islamicized. The Christianization of Anglo-Saxon Britain is in doubt, with missionaries coming only from Christian Ireland and the Brittons in the west. Whether Britain too would eventually become Muslim, or if the Anglo-Saxon Gods could hold on is anyone's guess.
In the East, Christianity reigns supreme in Egypt, in Anatolia, and in the Balkans, as well as in Ethiopia, which was the first country to convert to Christianity. The Arabian Peninsula is still a Muslim holdout, and likely to remain one, so long as the western Muslims continue the Hajj to Mecca and Medina. In Persia the state religion is still Zoroastrianism, though Manichaeism (a new religion drawing from Zoroastrianism, Christianity, and Buddhism) could supplant Ahura Mazda.
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So that is that. Sorry about the lapse in posting; I've been quite busy. Expect the next Alternate History Podcast sometime this week.