Many JW's will not know how to respond to this information when shown. The usual reaction will consist of immediate denial and will never want to discuss this issue again. In fact, many JW's will end all contact with you at this point as they will suspect that you have been looking at "apostate literature," or worse, talking to apostates themselves. But the honest JW will take this information and investigate it. However, their search will very quickly end in vain when they realize that there is nothing in the Watchtower's literature (in particular, the CD Library) that addresses the accusation. Thus, in an act of desperation they will use the trusty google search engine to find out if any JW's online know anything about this. Here is what a google search of "Watchtower United Nations" will pull up:

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If the JW is looking carefully, he will immediately disregard the first hit as an "apostate website," as the first thing one sees is the Watchtower riding the scarlet beast. However, if the JW is open minded, he will notice that the "Randy TV" website is filled with full documentation on the issue including official U.N. letters and scans.
The second hit contains the U.N. public letter in addressing the extent of the WT's involvement. The third hit is another "apostate" website that the JW will avoid. The fourth hit is a Wikipedia article that contains a basic overview of the Watchtower's history of views with the U.N. And if the JW hasn't given up already, he will find the final answer; the be-all-end-all to this "silly accusation" that will set "opposers" straight once-and-for-all. It is the Jehovah's Judgment Website. No one knows exactly who authored this website, which is unsurprising since the Watchtower discourages JW's from creating unofficial websites like these. But the JW can now rest at ease, because there is an "answer" to the accusation. It doesn't really matter to the JW if the answer is legitimate; he just cares that it is in depth and "addresses" the full scope of the issue.
The interesting thing is, JW's on the internet will immediately point you to this website. But JW's in person will almost never do this. The reason being, JW's on the internet are anonymous 99 percent of the time. And there will be no way for the elders to find out that they are using non-WT literature or websites in answering questions about the organization. In my experience, JW's who you meet in person will very rarely even bring this issue up again and will especially not point you to this website for the reason mentioned above. The website will simply set their mind at ease and they will likely never ponder the issue ever again.
But what if they do point you to this website? How do you respond? In my experience, it has done no good to provide any sort of "refutation" of the issue. To them, its already settled and the website has fully refuted the accusation. And unfortunately, this website contains the latest argument of the issue. That is, practically all websites, like Randy TV, which address this issue were written before the Jehovah's Judgment website. Thus, there are no specific refutations of the website (though the actual documentation of the issue speaks for itself). But now, someone has finally taken this website to task:
I haven't yet checked out all of the information on this website (I plan on doing so in time), so it is not recommended that you show this to JW's until you have ensured that the information and arguments are solid (though I have no reason to doubt that they are, as I personally know the author). The website can be found HERE. So, the next time a JW tells you to check out THIS WEBSITE to find out all the "answers," you can tell him, "Thanks for the link, but that website has already been debunked by THIS ONE. But we can go all day in passing websites back and forth. So why don't we just look at the documents themselves and discuss them?"
As of right now, I am in the process (actually, in the final editing stages) of an in-depth paper on the subject. Though there are more than enough websites that address this issue, I believe that my paper will provide a fresh look at the issue and will be written with the Jehovah's Judgment website in mind so that the arguments are as solid as possible and based on the best available reasons.