US will go forward no matter what, with or without official UN support. Though less countries would be inclined to commit forces reducing the overall manpower of the effort.
Seeing as thet made the bulk of the UN army, I don't see the early stages (Inchon Invasion and all) going different from OTL.
What you may see, however, is small expeditionary detachments sent from Taiwan to fight on the Peninsula to help bolster the manpower.
This of course could potentially butterfly into a more hyper aggressive PRC surprise attack before said troops get as close as they did to the Yalu in OTL.
With a more aggressive and earlier PRC attack in say late summer or early fall, you might see the US forces with less comparable manpower compared to OTL actually get swept off the Peninsula as well as US more inclined to use Nukes to save face.