What exactly makes a religion "weak" or "strong"? Christianity was absolutely mixed with ancient beliefs that formed from thousands of years of Jewish tradition, and as another poster pointed out, the very Greco-Roman philosophy that people were supposedly becoming disillusioned with. There were many social and cultural reasons why Christianity spread so quickly, but it was only the patronage of the Empire itself that made it the sole "legitimate" religion on the continent for so long. Paganism was ultimately stomped out by force and violence, not by some pure ideals that Christianity held a monopoly on.
As for Islam, I doubt it would exist in this timeline since so much of history would be different. Even if we say it does still develop, however, its spread will be very different. I would argue that the religion would actually be less successful, since it wouldn't be able to play off of the sectarian conflicts within Christianity. Many Christians in North Africa and the Middle East welcomed the Arab invaders in. The Church in Rome will torture or kill you for heresy, but the new Muslim overlords won't care as long as you pay your taxes. Those divisions won't exist in a world where Dodekatheism won out in the Mediterranean.
1. 'Paganism' may not have been replaced simply by persuasion by Christendom, however, the creation of Christendom was an evolution from the Mediterranean world prior to that of the new; exactly as the Middle Ages of Europe was in essence an evolution yet continuation of what Rome had created. It is not as if there is a Pagan Greco-Roman World which is then cut completely from the Christendom of the Latin world, rather simply an evolved one. Christianity adopted and borrowed enormously from the Hellenic Philosophy and we may say thus, that it is to a large degree, an Hellenic religion, in the same way that Manichaeism is Hellenic. Through this, we see that Christianity adopted not only from cultural characteristics of the Hellenes, but also its philosophy, especially that of the supposed Greek sages such as Aristotle and Plato, who even in their day had great critiques of the prevailing religious attitude of their day; Christianity would take the mantle of these ancient philosophers in 'renewing' the Mediterranean in its image.
2. If we assume that Christianity is relegated to minority in Judea and the east-south, the consequences for Arabia are not really known. For instance, if the Western Empire does not fall, then the consequences for this is unforeseen. As well, without Roman conversion, do we assume that Aksum still converts? If so, then we can imagine Himyar and Aksum still engaging in war over the Aden straits and the reverberation into the Hijaz, affecting the life of Muhammad's (SAW) parent's lives and creating the chance for Islam along with what had already been built up. We may say that within the Arab traditional religion, there was already a sort of concept of monotheism. The idea was that the gods separately represented certain aspects of the single deity and were intermediaries and this is what Islam in its early days called the Arabs away from, hence it was not dependent per sé upon a Christian Rome.
3. In the initial conquest of the former Roman Empire, by the Caliphate, the different sects of Christianity did not rebel against the Byzantines, rather they rebelled against the Muslim at Constantinople. Later there would be Gnostic 'Christian' representation int he Muslim ranks to wage war against Byzantium through the Paulicians, but otherwise, there was no sort of alliance between anti Byzantine Christians with Muslims. At least not to the extent that the Sassanids remained close to the Nestorian Church of the East as a sort of anti Constantinople alliance.
-Thus we may say, the victory Islam gained, was primarily a military victory. The Byzantines were defeated in the field and thereafter, were savaged in Anatolia and besieged twice by the Umayyad despite internal rebellion and dispute within the Islamic world. Without Christendom, I do not see how a haphazard pagan East can do any better than the Byzantines did otl.