If indeed we posit that Theodosius I transforms the Olympics into a Christian event sometime in the latter 4th century AD, then it's plausible at least that his successors in the Western Roman Empire might try to do something similar (perhaps as a means to better integrate the various nationalities flowing into the Western Roman Empire). Though the "Western" Olympic Games won't be nearly as continuous as the ones held in Greece, they stand a good chance of surviving the fall of Byzantium.
Perhaps the Western Olympic Games become the realm of the Papacy, held in order to maintain good relations within Christendom...Rival Eastern/Western Olympic Games probably persist until the fall of the Byzantine Empire. With the Fall of the Byzantine Empire, the Eastern Olympic Games fall dormant (though the western ones continue) until the Ottoman Empire sees the utility in such games. Under Ottoman rule, the Eastern Olympic Games are transformed into a multi-faith event intended to integrate the various disparate parts of the Ottoman Empire.
Following the Protestant Reformation, Protestant athletes, unable to participate in the Catholic Western Olympic Games, in a shocking move, opt to participate in the Ottoman Olympic Games instead, deliberately thumbing their noses at the Papacy. Over the course of the next few centuries the Western Olympic Games continue to lose legitimacy as fewer and fewer athletes attend.