The big problem with the in-game UN (problems with the United Nations OUTSIDE the game are another matter entirely) is that while it's built by ONE player, it really isn't meant to benefit that one player. The UN is supposed to be a benefit to everybody.
I have no problem with only one UN-style wonder. Past organizations such as the LoN, and others before that, are still based on the same basic concept, and a single in-game building can be used to represent them all.
The UN wonder needs to be remodeled so that it gives a worldwide-benefit impression, while still giving one player a reason to build it. Either that, or change the UN into something outside the scope of wonders, such as the Orion Senate in the Master of Orion games, which is created automatically once a certain percentage of all the planets on the starmap have been colonized.
A few possibilities....
1: The player building the UN receives extra happy points for being the world's center of diplomacy.
2: The UN gives extra happy points to EVERYBODY.
3: The UN raises war weariness for everybody (i.e. worth building if the player wishes to AVOID war and gain a Space Race victory or culture win, and minimize interference from the AI's)
4: The player who builds the UN automatically gets a seat on the Security Council; anyone who wishes to join the SC can then do so. If anyone in the SC is attacked, all other members of the SC are REQUIRED to declare war against the aggressor; civs outside the SC are not protected by this process or required to declare war. Therefore joining the SC has pluses AND minuses.
For extra realism, members of the SC could be required to pay some gold per turn, and "donate" military units every now and then.....