I will come to the religious changes, later. The thing is, to avoid this, there needs to be an Economic and Technological Renaissance in the Byzantine and Sassanid Empires that actually increases the population of the region in their empires that ALSO causes large Social, Institutional, Intellectual and other changes, in which case the Arabs aren't dealing with the same two empires of the OTL. This is because the main factor in the Arab conquests of the Middle East wasn't just Islam, which was just a culmination of a variety of social movements that were happening in that society, but a population explosion in that region likely caused by the fact that these weren't touched by the Political instability , Plagues and Wars, to the same extent as the empires, and also the fact that there was a Climatic shift in those ages, which caused a havoc in Europe, but might not have touched the Arab regions.
Let's just say that the Byzantine and Sassanid Empires DID manage to do the above and survive. The reasonable change would be that the Western Levant would be Aramaic and Canaanite speaking (latter in pockets), while having a large Greek speaking ethnic minority, or even a Greek proportion nearing 50% due to a variety of reasons. Then you might have a presence of other Eastern Orthodox/Byzantine Catholic languages like South and East Slavic, Armenian, Georgian, Caucasian (who are all Eastern Orthodox/Byzantine Catholic (ByzCat here on) in this timeline), in scatterings, because an empire consolidating power so much would indeed have a lot of intertwined relations with neighbors.
Come to the Eastern Levant: Like say, Syrian Desert and Jordan (outside the Greek dominated North and a few Canaanite speaking cities on the Edge of the valley). These would be majority North Arabic speaking. However, not the Hijazi Arabic of the Modern World. They would be Arab Christians.
Move into Mesopotamia: Arabs would be an important and a large part of the population mix which then also has a huge mix of Persians, Kurds, Jews, various Turkic peoples, Gypsy and Gypsy-like tribes, Arameans/Mesopotamians, etc. Armenians and Caucasians have a small presence, too. This region is under the Sassanid Empire or it's successor until the Modern Age.
Religion in the Levant would obviously be overwhelmingly Christian, but we cannot comment about the sects/Churches/etc. Arabs would be overwhelmingly Christian in the North, with Pagan Arabist cults beginning from Jordan to the South, into Hejaz, where of course, Judaism and Christianity would also be a part of. Arab Paganism doesn't die off, because their role in the trade in a continuing Antiquity, and the Sassanid Buffer would help them. Further into Arabian Peninsula would see various diverse tribal cults incorporating elements of Christianity, Judaism, Zoroastrianism, etc.
Religion in Mesopotamia would be very complex. Persians would be Zoroastrian of course. The Arabs would be a mix of Pagan, Christian, Zoroastrian and other. Judaism would have an important presence in the cities. Pagan and Native cults could be prevalent among the Assyrians/Semitic Mesopotamians and similar with the Kurds. Hinduism will indeed have a small presence, mainly followed by the Gypsies, Jatts, Sindhis and other ethnic groups that come under the empire. Buddhism could be prevalent among the Turkic peoples along with Tengrism and Zoroastrianism. All this assuming that no large scale religious movement happens in the Sassanid Empire. Also, there will be other Iranian peoples of the North and the Northeast, not just Persians and Kurds. And of course, since the empire has a hold on Sogdia and Bactria, you will likely have a small scattering of those people, too, who likely follow Buddhism. Nestorianism, all the while, is prevalent among all ethnicities, to an extent.