Presently it could reasonably be said the Islam is Christianity's only meaningful rival(other then secularism of course). Adherents of Dharmic religion, while numerous, have little drive to proselytize(not withstanding the past drive by Buddhism which seems to have died down) and seem inherently inclined to losing ground to Abrahamic faiths.
Considering this, I wonder how hegemonic Christianity might have been if Muhammad had never been born. Christianity had been gaining ground in Arabia(albeit surprisingly slowly compared to it's rapid rise in other "primitive" pagan societies), albeit of less conventional forms, so it seems likely that Arabia would eventually succumb completely to Christianity. As for Persia the state was an active proponent of Zoroastrianism but I have some doubt whether it would endure past the regimes inevitable fall- Christianity had a large presence in Persian territory(dominant in Mesompotamia), and Christianity was spreading among Central Asian who would likely conquer Persia in an ATL as in OTL. And given the typical attitude of the era's Christians to religious minorities I doubt the Zoroastrians would survive at all like they did under Muslims.
With Persia, Arabia and Central Asia Christian, Christians would control the trade routes between East and West. It does not seem unreasonable to suspect that Christian merchants would have successfully proselytized the The SE Asian archipelago with success similar to the Muslims, nor to suspect that Christian tribes from Central Asia would ultimately conquer India- ruled by Christians Hinduism would surely fair worse in India then it did under Islam.
And with that the only significant redoubt against Christianity in the Old World would be East Asia.