In my understanding there were material reasons why Rome adopted Christianity, according to some theories it was a conscious decision on the part of the Roman ruling class seperate from Constantine.Roman peasants , once the military backbone of the republic, had been dispossessed, in either living in semi serfdom on latifundia or packed into teeming cities, entertained by the famous "bread and circuses". The empire was dangerously over extended, but yet had to continually expand to increase the supply of slaves and tax revenue.Christianity, by offering salvation to downtrodden masses, could tie them to Rome while continuing elements of the Roman state. Christianity , which despite being officially a continuation of Judaism, was an amalgamation of several pagan traditions. According to the Pagan Christ by Tom Harpur, Egyptian immigrants in Rome, a hundred years before Jesus was supposed to have been born, worshiped "Mary Mother of God", as part of a syncretic Egyptian folk religion. Possibly with no Christianity something very similar to it might have emerged. Maybe not though. whatever emerged might be more "pagany", without the Abrahamic absolutism.
An uneducated hot take. The Roman elite during the late Empire were already pretty much atheist or agnostic. Without Christianity some form of neo-platonism becomes popular with intellectuals. Stoicism continues. In various parts of the Empire saviour cults emerge. These have many local variations-some are militant revolutionary movements, others preach quietism, "render unto Ceasar what is Ceasar's, others are fiercely austere and are similar to Jainism. These are generally considered to be part of the same movement.Perha[s a severly austere religion catches on among the aristocracy in Egypt (in OTL monasticism was huge in late Roman Egypt) The religions in GOT reflect this, I think. Elements of Buddhism , Hinduism, and yoga traditions enter Europe, although there aren't major conversions.There are fusions of Greco-Roman mystery cults combined with Mithraism and Egpytian ideas.None of these become a major religion however.
The Empire collapses about the same time it did in OTL. The Dar Ages either lasts longer, without Christian monks copying manuscripts, or is much shorter, without the break in Hellenistic Classical culture.The Empire is chopped up by Germanic tribes as in OTL. Paganism continues but politics and the development of west European nation-states doesnt revolve around religion or churches. Nation states themselves might develop differently, perhaps Europe would remain an amalgamation of statelets, tribal confederacies and trading leagues. Temples dedicated to one god or another might compete or favor rival leaders, but there wouldn't be anything lie the Reformation or the Thirty Years war. Eventually some temples, say the temple of Wotan Allfather in London, begin to sponsor recovery of Classical learning and later the scientific method, although there are those, like Zizek who feel that Christianity's anti pagan break from nature was necessary for the scientific revolution.
Sorry for the rant.