First of all, I apoligize in advance for the immense post. It's just because I was actually thinking (and researching) about this PoD through this last months, and am happy to suddenly see it posted

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Ok, from what I've found out, Constantine I already had Christian sympaties even before Milvian Brigde (possibly due to the fact that his mother was already a devoted Christian), while neither Maxentius nor Licinius had a similar regard, so it's highly unlikely any of them will convert (neither would Galerius, if still alive).
From this observation, in the micro-historical aspect, I imagined two opposing outcomes:
(1) a single-minded Pagan autocrat renews the persecutions carried out during Diocletian's reign under Galerius (even if the religion had been legalized since 311 by the Edict of Toleration), perhaps to strenghten the Pagan institutions as a symbol of Imperial power (e.g. forcing Christians to offer sacrifice under pain of death). I doubt this will actually extinguish Christianity, but, depending on the scale will delay its
momentum.
(2) Some later Emperor will effectively legalize it (similar to the Edict of Milan in 313). Either it will stop the bloodier phase of persecutions or (if antipathetic to the Christians) simply will bar non-pagans from certain offices (like Julian the Apostate tried to do). Even if this doesn't stops the Christian religion from expanding, it will reinforce their status as a peculiar cult, like Judaism.
That being said, it's arguable that the Constantinian dynasty's association with Christianity was a political move to reinforce the central authority of the Emperor... so, a likeminded Pagan sovereign will make serious efforts to structure the Pagan institutions as a mark of the Imperial rule, as Aurelian seemed to do, or Diocletian by his vinculation to Jupiter. After all, both Pagan and Christian Emperors were keen to claim legitimacy from the divine support.
I find it reasonable to think this renewed central religion will be emphasized on the Cult of Sol Invictus (as a Roman god, independent from its Syrian origins), with a possible effort to blend it with Mithraism - even if the cult of Mithras is a mystery religion, the official Imperial Cult will create fictious links between the beliefs (easier said than done, of course, but both Sol and Mithras were solar deities) to guarantee the adhesion of the military as a whole to this New Imperial/Solar Cult. I guess this strong Pagan Emperor might want to be blend the Cult of Isis as well, due to its popularity with the female population (or, better, with the Eastern peoples as a whole). If even OTL Christianity allegedly has some inspiration from Isis' cult, I'm sure a Roman Pagan system would try to integrate it.
The overall result is a mitigated polytheism. I don't believe it will become, at least not for a few centuries, a pure monotheist religion, but perhaps evolves into a flexible form of henotheism (worship of a single god while accepting the existence of other deities). I doubt it comes close to what Julian the Apostate tried, because he was much more influenced by the traditional Hellenic ideas, and had a philosophical inclination. Military-minded autocrats like Aurelian, Diocletian (and even Constantine himself) adopted religion not as a philosophy or a way of life, but as a means of ensuring the Imperial domination of the wordly affairs, promoting political, cultural and social cohesion to stabilize the Empire. Any ideas of divinity served to ensure the legitimacy of the Roman Emperor, so I find it probable that will stress the mainline Roman traditions instead of, for example, the archaic Greek, Syrian, Egyptian ones. Also, it will try to become more homogeneous, to promote unity, in opposition to the Classical Paganism which assimilated alien deities to their own pantheon.
On the effects on Christianity itself, I honestly doubt it prevents its internal divisions about theological inconsistencies. Perhaps it doesn't comes to a serious schism like Arianism or Nestorianism, but I hardly believe that Christianity will be (as seem from a Christian perspective) homogeneous. It's possible, regarding its neutral influence on the Roman affairs, that Christians won't adopt some Pagan features, and will try to distinguish themselves from the "Pagan culture" - like the Jews did since the diaspora, insulating themselves from every foreign culture. As a disclaimer, I doubt Christianity will see a similar development as OTL Judaism, simply because it was much more heterogeneous and had a number of adepts, while Jews were always a minority.
Another point I find extremely relevant was that Christianity by Constantine's reign was much more entreched on the eastern side of the Empire, and had a significant presence in the metropolei of the western half (Carthage, Milan, Rome itself), but Paganism was predominant. If the Empire experiences another division (either on purpose like Diocletian/Maximin or enforced, like Honorius/Arcadius), its much more probable, unless butterflies say otherwise, that the Eastern Empire becomes Christian (either a pagan Emperor is baptized or from an already Christian situation rises to power), while the Western Empire possibly remains Pagan. If the halves of the Empire eventually become hostile to each other, its reasonable to imagine that the religious differences will be emphasized (i.e. Christian vs. Pagan)
[Disclaimer: I don't follow the historiographical tradition that helds the adoption of Christianity as the cause of the fall of the Western Roman Empire. In this ATL Scenario, the fortunes of a Pagan Empire are still linked to the ripples of the Migration period. The Huns will come sooner or later, and the Germanic peoples will be inevitably pressed against the Roman borders].
Finally, regarding the Barbarian kingdoms, its out of question that they converted only because of the Roman sphere of influence, in a domino effect (just like the Bavarians converted under the Frankish suzerainty or the Bulgarians due to Byzantine influence). If I recall correctly, the first Germanic people to convert
en masse were the Goths, by the late 4th Century, and eventually the Burgundians, Alamanni and etc.
In this "pagan" scenario, depending on the extent of the Christian discrimination by the Pagan authorities, I can fancy some outcomes:
(A) The Christians will be slowly driven to the fringes of the Roman world (a parallel, IOTL can be made with the Nestorians and Pelagians, after they were branded heretics, formed religious strongholds in the very frontiers of the Empire), and might come to contact with the Germanic/Iranian peoples. The Christian doctrine might be popular with lesser castes, but it's possible the Barbarian aristocracy will see little interest in abandoning their traditional beliefs to adopt a foreign religion that doesn't interests the Roman rulers. The conversion of a handful of tribal rulers will be the exception, rather than the rule, due to the more concentrated efforts of persuasive missionaries.
(B) On the other hand, the Barbarian rulers might be somewhat interested in Christianity (depending on the adhesion of their subjects) if it represents a symbolic opposition against the Roman Empire (like the Hungarians adopted Catholicism to avoid falling into the influence of the Byzantine/Bulgarian Orthodoxy). This is a fascinating scenario, imho, as TTL Christianity might become more Germanic-Pagan influenced than Mediterranean-Pagan. This, in turn, might jolt the Roman Empire to invest in the convertion of the tribes under their influence to the New Pagan/Solar Cult.
(C) Perhaps the Christian or Roman-Pagan influence upon the Barbarians is negligible (I find it more likely, because, unless due to serious political efforts, polytheism is not inclined to proselitism, due to its extremely incoherent nature). So this will be an alien world, with a marked division between the "civilized" Mediterranean/Hellenic-Pagan world, the Germanic bloc on the other side of the borders, with the eventual arrival of the Slavs and Turkic peoples to bolster the melting pot of Paganism in Europe
