We are talking about almost a thousand years worth of butterfly effects from this POD so anything that could reasonably happen could happen. I can talk about some macro-scale things in certain regions but smaller things like what peoples or dynasties would rule and where are completely beyond me or anyone, really. It's all up to the writer of a timeline.
Brief declaimer though, the Turkic migrations are going to happen. Butterflying Islam away doesn't change that. A nomadic empire showing up in the region at least once is almost a certainty and can throw everything I'm about to say into chaos, especially in Persia.
A brief refresher on early Christianity because it is very important to what I have to say. Christianity was divided into five episcopal sees called the Pentarchy, the capitals being Rome, Constantinople for Eastern Rome, Alexandria for Egypt, Antioch for Syria, and of course Jerusalem. This was a resilient system and even under Islamic rule it endured, so I expect it to play a massive role in the region without Islam. These centers of Christianity will be jockeying for influence and will be trying to subordinate each other. What does this look like in practice? Lots of war and nation building. Christians in TTL may consider themselves as a sort of family, however they will see the different types of Christianity as almost as different religions. Even in OTL Christians have not hesitated to spill blood over semantic differences and it would be even worse here.
Constantinople/Rome/Byzantium: What else is there to say? The Empire of the region. The second most powerful See in Christianity, powerful enough that it could challenge the Papacy itself at times. Would be by far the most important state in the region. It also has very real institutional issues because the emperor is so powerful that regicide is very lucrative way to get the job. Plus to this is upward mobility, downside is political instability. The systematic issues the Empire suffered in OTL would be just as bad in TTL, or even worse because of the lack of the existential threat the Caliphate posed to both the State and its religion. It is the maker of history here, and what will happen to the region would be a consequence of what would happen in times when it is weak or reasserting its influence when it is strong .
Alexandria/Egypt: This is an interesting one. Alexandria is far enough away from Rome and Constantinople that it will be able to assert it's independence and this has ramifications on Egypt. Egypt had been under Greek rule for several centuries at this point and the upper class was thoroughly Hellenized. Coptic was still an important language though. I'd say there is a almost 50/50 chance that Coptic would reassert itself as the language of Egypt once more as a form of Copt reactionary movement against Constantinople, Rome, or both. The other 50% is Egypt becoming Greek. Greeks had ruled for several centuries after all, what's stopping them from ruling for several centuries more? In that case, just like how Egypt in our OTL was Arabized, in TTL would be Hellenized. I will stress this does not mean the Egyptian Identity disappears though. Egyptians could be culturally Greek, speak Greek, and still think of themselves as Egyptians first. I think it would also all but guarantee Greek would split into multiple languages, like what happened to Latin and Chinese.
Antioch/Syria: Another interesting one . Think of this one as a clash between the Hellenized Anatolian Greeks and the Semitic inhabitants of the Levant and Mesopotamia. Constantinople can assert as much authority over either region as it wants, and while some Hellenization will happen, this region will stay mostly Semitic. Over time I think kingdoms and nations could gain their independence in this region as a type of cultural backlash against Hellenization.
Jerusalem: The one everyone will be fighting over. Constantinople might try annexing it but when one of it's many civil wars happen over the regicide of the month, I think it would be tempting for Rome to try to assert its influence here. There would be a lot of pilgrims, much more than OTL. While pilgrimages are not mandated in Christianity like they are in Islam, the City of Cities being under Christian control means there would be centuries to develop a culture of pilgrimages. Jews would probably not be welcome. Christians would consider Jerusalem to be their city and would resort to violent means to keep it.
Persia: This is the real wildcard. After the Great War of 602-628 the Sassanid's were screwed. The massive political chaos that occurred is only what I could describe as the end of a dynasty. The Sasanians will not be sticking around, but who is going to replace them? It could be a cadet branch, a Persian House, or a nomadic invader. All of these are plausible, and all of these could plausibly follow after that dynasty falls. Zoroastrianism could stick around, it will need make reforms though so that it could conceivably attract and convert Christians to this Neo-Zoroastrianism. Or Iran could become Christian itself over time, with modifications made to make it attractive to Zoroastrians. This Persianate Christianity could end up pretty weird and be considered heretical by Nicaean Christianity. Buddhism was popular in Eastern Persia, perhaps it could have been spread West and Neo-Zoroastrianism is a weird syntheses of Christianity and Buddhism. I can't say much more, Persia is a blank slate.
Arabia: Goes Christian with almost 100% certainty. Even before Muhammad's time, Christianity had made it as far south as Yemen and crossed over into Abyssinia. The pagans would stick around for a couple of generations and may end up adopting some weird forms of Christianity that could stick around for a while. Arabia does lend itself well to tribalism. As for the Arab tribe's themselves, some would travel North like OTL. However they would end up assimilating into where they settle down and Arabization would be pretty minor.