There would be no economic power to back up a UN currency, meaning it would be dependent on voluntary participation from the bulk of the member states’ economies, which likely means that it would quickly devolve to either a protest currency used by anti-west regimes, a slightly federated version of the dollar that is responsive to the needs and desires of mainly the US and partially the EU, or it will be dropped/ignored by both the West and the Anti-west and become a currency of minimal value that is used only on the fringes of the world economy. The UN simply does not have the centralized capacity to operate a currency and enforce that currency’s use amongst it’s member states, especially those that already have a hegemony that would be threatened by such a currency

Largely this doesn’t happen because the US dollar already serves as a good enough reserve currency. There’s always been reserve currencies and unless the US abuses this power too much, there isn’t any real demand for an alternative.

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That’s not really how currencies work. They are not just something arbitrary, they are a thing people trade which has value formed by supply and demand. People buy dollars to be able to buy things that are sold in dollars. Same goes for other currencies.

What would be the demand for UN currency? What can you buy with it? How would the price be determined?

The idea that USA somehow hugely benefits from having so much international trade done in dollars is also a bit weak. It does give them some international clout but that’s about it. There are some very complicated things relating to trade balances involved when your currency is the global reserve currency.