Well, I will try to be clearer, sorry for long message:
You just forgot eight centuries of butterflies. Beside I don’t think that ancient philosophy would have died without Christianity and to be honest this “ Christian monotheistic appeal” can be attributed to the fact that basically the most powerful man in the empire was always a Christian since Constantine, thus conversion to the new religion made the career easier for lots of ambitious opportunistic guys, otherwise Christianity would have remained what it was: the religion of a minority quite influential but not powerful enough to overwhelm the Pagans.
Butterflies are important but I usually prefer concentrate my attention on historical facts and their consequences: too butterflies can transform a good TL in a fantasy work and it's not possible make a discussion about fantasy works. When I wrote that, without Christianity, Ancient Philosophy would be die, I meant that, without the monks who copied almost all old ancient texts and the monasteries that protected them during Middle Age (and Christian Philosophy that partially included Classic Culture), probably there will be not more many philosophical texts and you cannot learn philosophy without texts. You can spread a simple religious or political message without texts, but you cannot teach all the Plato's works without his texts. Without Christianity, there would be very few people able to understand a text about it and no one interested to do it.
And why Constantine decided to convert? Because he had a dream? To begin the only conversion traced in History is a supposed deathbed conversion and not to Catholic Christianity but to heretic Aryanism. But let's pretend Constantin converts for his reasons: why all the Emperors after him decided to do the same? Many dreams? Let's try to be serious: many Emperors and senators, maybe Constanton himself, probably choose to convert for convenience, of course, but this was convenient (and this is the point) because Christianity was a young appealing religion in great expansion. In 300 approximately a tenth of world population was Christian (source: Wikipedia, Christianity in the 3rd Century). The man in charge was Diocletian, a infamous enemy of Christians, so it's clear that the incredible expansion of Christianity (because in the 3rd Century, with the means of this age and persecuted by the most powerful Empire of the moment, converting a tenth of world population in less then three centuries is a huge success, no discussion) is not, at least at the beginning, linked with collaboration with Roman autorities. Second the same source, in 280 (Emperor in charge was Probus and I doubt strongly about existence of so many "establishment-opportunistic career" men in rural zones) Christianity started to spread in rural areas, where Pagan traditions lasted for centuries but began quickly to deteriorate, disappearing or, where possible, integrating with Christianity. According to population estimates, in 300 world population was around 240 millions and Europe was 15% of total, so 36 millions, plus some millions of inhabitants from North Africa and Middle East and less some one from North Europe, let's say 50 millions of people lived in Knowed World-Central and Southern Europe, Mediterranean Basin and Middle East-, where Christians were concentrated: that means that the more or less 24 millions of Christians in 300 (10% of 240 millions) were almost half (!) of Roman Empire's and near countries's total population. An other important thing: Christianity is an organized religion with a powerful social and personal message, Paganism was a bunch of cults where the main part was sacrificing animals to satisfied some bellicose nature-gods. Christianity has organized institutions and its message, its cultural background, is able to quickly mobilized a large group of people, believers and not believers. Paganism didn't have nothing to similar and saying something about a ripped cow usually don't have a great appeal. So, when I spoke about "Christian Monotheistic Appeal" I don't speak about religion itself- I don't mean a divine plan or something similar- but I speak about the power of Christian Word, not in religious meaning, but as the force of a preaching and of a message that converted so much people in so few centuries, despite massive persecutions and strong resistences.
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.
Well, I'm not questioning the "force" of any religion but we can not fail to consider the ability to drag the crowds: for its social message and, yes, also for its religious message, more centered on the single person, forgiveness, fight against human sufferance, brotherhood and others good sentiments and inner experiences Christianity was and actually is able to answer to human doubts more effectively then Paganism. As I said before, we have not to consider specifically religion itself but their ability to convert and drag and in this Christianity was decisively in advantage.
Christianity is not only mixed but strongly rooted in Judaism but: 1) many of more restrictive Jewish traditions were overcome by new Christian one, not randomly, but for explicit teaching of Jesus (for example about food) 2) no one in Europe knew a damn about Jewish traditions so this can not have influence about Europeans conversions. If you read what I wrote, you could note that I didn't wrote "philosophy" but "beliefs": people were disillusioned about their archaic beliefs, not about a philosophy knowed only by few studiouses and that was saved from invasions and devastations by Christian monks and partially integrated with great success in Christian culture.
The Empire legimitated Christianity? Really? The Empire tried to kill more Christians possible until it could. When Constantin conceded religious freedom with Milan Edict, considering numbers of Christian population, this was more a mandatory act: the Empire couldn't survive without support of its Christian part so wean say that was Christianity to "accept" the Empire.
Oh yeah, and do you want speak about violence? After three centuries of mass murderes and massacres, people ripped by lions, crucified, beheaded, burned and stoned, kangaroo courts and families forced to hid themselves in catacombs? And, despite centuries of violence, Christianity was able to spread in all the Empire, becoming almost majority when Constantin decided to legalize all religions. When Emperor Theodosius (the Emperor, eh, not the Church, the Bishops or any others) decided to declare Christianity State Religion in 380 and to prohibit sacrifices in 392 Paganism was a declining minority and Theodosius wanted to compact the community before the final crisi of Roman Empire. Violence occurred as some Pagans tried to revolt but it was largely a political act of the Emperor, not a religious one. While in many cases Christians participated to Paganism repression, in many others cases Christians tried to offer an act of mediation (famously when Milan Archbishop Ambrose strongly condemned Theodosius for Thessalonica Massacre). Christianity never had a monopoly of pure sentiments and surely acts of violence were committed by Christians during the last phase of Empire, but saying that all the process of Christianity's enlargement was based on violent conversion is more then reductive, is an historical error, that fails to explain as Christianity managed to survive and expand when violence was used against it and before its "legalization".
What about Islam? I said that butterflies are important but not so much. Speaking about other religions influences in Islam, Jews were more numerous then Christians in Arabia, so I can imagine a similar Monotheistic influence in Islam, as Judaism influenced it more then Christianity (Human-Divine Relationship, prevalence of Sacred Texts on Word, no Saints, role of Prophets, no celibate for Rabbis and Imams...). In 7th Century Inquisition didn't exist and persecutions against heretics were rare: simply the Church was not enough organized to enact such harsh misures. Sometimes, where Christians and Heretics lived in the same city, they clashed and rioted, similar to clashes with Pagans, but only the Empire was able to suppress such communities. Aryanism, the main heresy of time, was limited to Central Europe and survived so long because some Emperors and Kings initially converted to it. In the same way, Iconoclastic Crisis would not have degenerated if some Emperors had not supported the Iconoclastic Movement, transforming a religious dispute in a political question. Nestorianism was isolated in India and Persia and his relationship with the Church was never violent, as they reunited during Middle Age in the Eastern Churches. So, when you speak about a bad Church that kills and deport in North Africa (the land of Saint Augustine, one of most important Fathers of the Christianity) and Middle East, I simply do not know what you're talking about. People of Syria, Palestine, Egypt and many others places welcomed Arabs because they were overtaxed by Byzantines for their wars against Persia (that then overtaxed his own people too).
To conclude, I don't want make a defense of Christianity or the Church and I don't want sustain that Polytheism and Paganism are "stupid" or something similar. Opinions are free and discussion is open. I proposed only a simple POD to realized the target: having a Greek-Roman Religion alive without Christianity to interfere. If you simply eliminate Christianity, German People will still destroy Roman Empire but, without the process of assimilation between Germans and "Romans" fundamentally mediated by the Church and conversions, before or later Europe will be divided between some Eastern invaders (for me, Islamic Calipphate and Asiatic Khanates), that will overrunned weak Europeans Kingdoms. My POD was simple: Rome (because Roman Religion cannot survive without Rome) avoids Eastern expansion and concentrates its forces on German border, maybe conquering Germany and resisting to invasions. Greece and its culture can survive better without Roman conquest. A sort of Nestorian Christianity will become an important religion in Persia but will be limited in this area and maybe in India and Central Asia. Islam could born in same way but will be limited to Arabia and maybe same zones of Africa and Indonesia. Europe will be dominate by a healthy Roman Empire, with Roman paganism as dominant religion, and this was the target.