Paganism in late Antiquity needed a unifying force but I don't think that force had to be Abrahamic. Paganism in China (Taoism) was subsumed and explained by Buddhism and so was paganism in Japan (Shinto) and Tibet (Bon pa). There's no reason why, absent a Christian emperor and imperial proscription of competing religions, paganism (or Mithraism or Sol Invictism) could not be reframed by Buddhism in the Roman Empire too if Buddhist monks were getting to Rome via the Silk Road. After the 500s when the Sui and Tang began to expand trade on the Silk Road once again if not sooner.
Of course lack of Constantine might not butterfly away Muhammad and Islam--which emerge directly out of Judiasm.
Nor would the lack of Constantine prevent non-Imperial Christianity from becoming popular in places. There was a very vibrant British Christianity in Late Antiquity until the Roman Church supplanted it, alleged to have been founded by Joseph of Arimathea.
One thing I think is certain, though. Without an Emperor and an Empire enforcing Christianity (except in Ethiopia and possiby Armenia), there would have been no basis for one standardized Church to call other Christian schools heretical and make that charge stick. Christianity would have likely been much more diverse and syncretistic. The Arians would not have died out. The Donatists and the Nestorians would not have died out. The Coptic Monophysites would have as much validty as the Augustinian Romans and Greeks. And none of these Churches (or Sol Invictism) woud have the power to force itself on Asatru (Odinist) Germans and Scandinavians. The only likely faith enforcers might be the Muslims or the Zoroastrians. We might even see a sort of Shiite Christianity based as people like Dan Brown and the Priere de Sion based around Jesus's bloodline through Bar Abbas (son of the father) as the Shia Muslims base their faith around the Sayyid bloodline of Muhammad's family and his nephew Ali.