I'd like to suggest an alternative to "Christianising" the Olympics - for them to become a tool of diplomacy. What you need is for a Roman emperor, maybe in the aftermath of the Imperial Crisis, to start inviting delegations from allies of the Roman empire - friendly Germanic and Arab tribes, Armenians when the kingdom is aligned with Rome, even Indian and Chinese delegates. This turns the Olympic game into a source of prestige and income for whoever holds Olympia, but also into a diplomatic tool - an unofficial meeting of 'world leaders' every 4 years, with high-ranking diplomats accompanying their local athletes hoping to curry favor with Rome, and to forge alliances amongst themselves. After the Roman Empire collapses, the Olympic games become a part of the Byzantine diplomatic arsenal - selectively inviting favored states or even factions within a state - for example, inviting Abbasid delegates to participate in an Olympic game during the third Fitna, to indicate a preference in a civil war. Olympia becomes the second city of the empire after Constantinople, and is oftentimes the wartime capital during the late empire when Constantinople is sometimes temporarily lost.
Upon the arrival of the Ottomans, (or any other Turkish/Arab/Slavic/Norman/Whatever state finally displaces the Byzantines), the Sultan embraces this ancient tradition as a way to legitimize himself, and of course keep this powerful diplomatic tool. The Olympic games remain almost unchanged from the Roman model, maybe with a slight Ottoman reflavoring, and as the modern age begins they become truly global - the United States are admitted into international politics when they are invited to the Olympic games of 1802.
With the establishment of the UN or an equivalent body, the management of the games are transferred into international hands, and they become an official global event, with all UN member states participating, not only in the athletic parts but in the international summit accompanying them.