I don't think there can be said to be a single "biggest loser" from the rise of Christianity. There are certainly things you can point to and say that Christianity's rise resulted in their diminishing: for example, the Mithraic rites popular among the military and bureaucracy of the empire, or if we want take a somewhat longer view, one could easily argue that Christian dominance has hardly done Jewish people many favors. You can even go a bit farther, and note that without Christianity it is highly unlikely that Islam would have arisen, or at least arisen in the way that it did. And Islam outcompeted Manicheaism in the Tarim Basim, essentially ended Buddhism in the western Steppe and almost entirely drove Zoroastrianism out of the Iranian Plateau. Of course, Islam, Christianity, Buddhism etc didn't actually do anything--ideologies only exist in the minds of people who hold to them. And you could also say that "Islam's" conquest of the Steppe was accomplished through the highly syncretic Sufi tradition. Anyway.

What I think is more interesting to do is to examine the world of religions that was irreparably altered by the rise of Christianity. And so instead of a single religion "defeated" by Christianity, we can talk about a general religious environment that was bent, but not broken, by Chirstendom. It is a little bit too simplistic to think of this as one of diversity collapsing into unity--temples to Augustus proliferated across the empire, and after the entire area was "officially" Christian there was a vast difference in rites and ritual (early medieval popes expressed dismay at the popularity of graveside feasting, for example). But it isn't entirely wrong. The religious world of the Mediterranean before the rise of Christianity was blisteringly diverse, with a vast array of local and universalist religious traditions. You might point to, for example, the omniprescene of symbols to Tanit in Tunisia for a good example of a widely felt but deeply local religious tradition. Or by contrast the Isis cult, which pretended to be Egyptian but in reality was something very innovative (and, at least if we follow the account of Apuleius, was highly tied to neo-Platonic cosmology).

Christianity didn't end this, it isn't as though after Constantine people from Gaul and people from Africa had the same religious style. But a big difference between Christianity and other cults (using the technical definition) of the time is that Christianity brooked no competitor. It was entirely possible to wake up thinking about Isis, go perform a Mithraic rite after breakfast then eat at a feast for Poseidon at dinner. If your god is the One True God and His son Jesus the Christ, however, this isn't quite as easily done. This is speaking through my own voice rather than any sort of scholarly consensus, but this unity, or consistency, or intolerance, however you want to frame it, meant that while early Christianity existed in a world of religious diversity, it didn't really have any real competitors.

There is a whole variety of works on Roman religion, Mary Beard's Religions of Rome is a great starting point, but for more something a bit more specific, Cliff Ando (ed) Roman Religion, particular Greg Woolf's article in it "Polis-Religion and its Alternatives" have a lot of influence on my thinking.

May I pose a question of clarification? Is it specifically Monotheistic Religions that you are wondering about?