I would say that the rumor goes all the way back to the expectations JWs had during WWII itself about what would happen in the post-war period. Rutherford taught that the totalitarian combine would get control of America and Britain and would forcibly abolish the governments in those countries, declaring "peace and security" by establishing a "new League of Nations" (cf. the agreement between the Axis Powers, Japan and Hungary on 11/20/1940 to establish a "new League of Nations"). Before that happens, however, the fascist totalitarians would have to entirely squelch the activity of JWs in those countries, as they had been doing in Germany and elsewhere. This is the final assault against "true religion" (a term that Rutherford himself, of course, rejected). Then at the moment the Axis Powers have complete victory, Jehovah and Jesus would step in and prevent them from destroying their servants.
That's what Rutherford taught in December 1941. Then he died the next month. But a week before he died, the Declaration by United Nations was promulgated by the Allied Powers. This suggested to the new president, Nathan Knorr, that the new League of Nations was already underway, that it was going to be called the United Nations, and that it was backed by the Allied Powers -- not the fascist "totalitarian combine". He thus dropped Rutherford's claim that Axis Powers would win the total war but maintained his expectation of a resurgent "League of Nations" that would declare peace and security, with the totalitarian powers playing a major postwar "religious" role. Drawing on his precedesor's interpretation of Revelation 17-18, but with the new United Nations in mind, Knorr prophesied that this postwar United Nations would attack religion:
"But mark this: The prophecy shows that when the 'beast' comes out of the abyss at the end of this total war it comes out with the woman 'Babylon' on its back, or she climbs upon its back as soon as it gets out. That means that 'organized religion', and this time the religious organization with headquarters at Vatican City, will ride and exercise guiding influence over the League beast...the United Nations opposing the totalitarian powers voice their pledge of freedom of religion as one of the four freedoms to be secured when peace comes....But what shall befall the 'woman Babylon', 'organized religion', riding the beast? At the end of that 'one hour' of the rulers with the beast, they will be forced to see the hypocrisy of religion and that it deceived them with false hopes and led them into a hopeless position with destruction as the only outcome...Will the religionists and all other persecutors of Jehovah's witnesses take note, please, that it is not Jehovah's faithful witnesses that will thus violently break up 'organized religion'? The world powers will do it, the very ones with whom religion is commiting spiritual fornication (Peace -- Can It Last, 1942, p. 22, 25-26).
"In September, 1939, total war broke out. Thereafter more than thirty members (the majority) of the League of nations became actively involved in the struggle between totalitarian and democratic political powers...Thereby the essential functions of the League of Nations and its Permanent Court of Arbitration were forced to suspend operations, and the seven-headed 'beast' took a fall down into the bottomless pit or abyss, but not to stay down. The prophecy says that the international 'collective security' beast will 'ascend out of the abyss' and 'shall be present'...This must come at the close of the total war and at the international effort then to restore peace and security. Even during the dark hours of the total war the revival of such a mechanism of international cooperation was widely and soberly discussed, especially among the United Nations, and an international police force was proposed to safeguard the peace and enforce the decrees of the 'beast'. Hence the beast must have horns, and the Revelation prophecy shows that it has 'ten', which is a complete all-embracing earthly number. This police force will encompass the earth, but particularly 'Christendom'...This beast has learned quite a few totalitarian tricks during the total war, and it will not unlearn them or discard them in the postwar 'new order'...With so many discordant selfish elements making up the postwar association of nations, Babylonish religion under the Vatican's leadership will service as the binding tie, but only for a time. The Vatican will act as a supranational power over the postwar confederacy of nations....In the postwar 'new order' Jerusalem's modern counterpart, 'Christendom', will surround herself with armies of an international police force for her continued world domination. The Roman Catholic Hierarchy will attempt to act as the spiritual police force of the entire earth. 'Christendom' will then feel she can cry 'Peace and safety'....'Christendom' will use her postwar 'new order' and her encompassing armies (Jesuitic as well as military) for the suppression of the Kingdom message. She will claim it is treasonable and disturbing and unsafe to the postwar rule...Because 'Christendom's' desolating abomination will then be standing presumptuously in the holy place God will cause desolation to be wreaked upon her. He will strike confusion into the enemy ranks. The hitherto docile political and commercial powers will awake to realize how organized religion has befooled them. They will take vengeance upon the religious organization" (The Truth Shall Make You Free, 1943, pp. 337-340, 342, 344, 349).
That is why the Watchtower teaching about the UN abolishing religion worldwide bears no resemblence to political realities. The teaching is a vestige of what the organization taught concerning what the postwar UN was supposed to be before the UN was established. It was supposed to be a (totalitarian) confederacy of nations under Vatican control that establishes a police force (consisting of real armies) over the whole earth. It was supposed to be a powerful military organization, a body that had the ability and power to "break up" Christendom. That is what the Watchtower prophesied the UN to be. And the destruction of the nations would shortly follow in the early postwar period, not over 60 years later. Although these are failed prophecies, the Society still claims success, because they can claim that Knorr predicted that the UN would be established before it came into existence. And so they continue to teach a version of Knorr's prophecy made back in 1942, even if it clashes with the rather impotent nature of the UN in the world today.