(This answers addresses the original question, which was more focused on whether the site was legitimate or was just made to be fodder for conspiracy theories. The current version focuses on whether it is affiliated with the UN, and see tim's answer for a better answer to that using the official word from the UN)
It's a real website and appears to be owned by a lobbyist/activist group which is focused on getting the UN to work on certain topics, but it is not directly associated with the UN itself.
According to the WHOIS database entry for unnwo.org, the site was created in Oct. 2018 and is registered to Jayme Illien of Illien Global Public Benefit Corporation. Illien himself doesn't seem to have a convenient Wikipedia page to link, but here's his Twitter page.
Although it's possible that someone not associated with Illien registered the domain, the content and style of unnwo.org seems to line up with the corporation's website, namely a focus on getting the UN to focus on happiness, and Illien's Twitter feed once referenced a 'new world order' initiative. However, after a quick look there doesn't seem to be any specific mention of the "United Nations New World Order Project" on the company's website, nor a reference to unnwo.org on his Twitter feed, so it's still possible that the website is not associated with Jayme Illien or his company.
Regardless of whether unnwo.org is owned by Illien or someone pretending to be him, it is almost certainly not a UN website. I haven't found any actual claim by Jayme Illien to be employed by the UN, nor anywhere on Illien Global Public Benefit Corporation's website saying they were founded/are controlled by the UN. Further, none of the terms 'Illien', 'New World Order Project', or 'New Economic Paradigm Project' seem to appear on quick looks through the UN's main website, https://www.un.org. While not an exhaustive search by any means, if unnwo.org was an actual UN website or Illien a employee of the UN then such a search should have returned something.
And as a final, if minor, piece of evidence, unnwo.org uses the wrong shade of blue: the line above the 'Initiatives' header uses #4e5ed0, rather than #009edb as defined in the UN's style guide.