The YouTube app for Android is starting to support HDR videos on select devices. Google is currently rolling out the new feature. You'll be able to see videos shot in HDR displayed with HDR in the app if you own a Google Pixel (or Pixel XL), Samsung Galaxy S8 (or S8+) or Note8, LG V30, or Sony's Xperia XZ Premium. That's the entire list at the moment.
As usual with Google's rollouts of new functions inside its mobile apps or services, not everyone who has one of those handsets is currently seeing the HDR support added.
In order to tell if you're one of the lucky ones, you need to play a video that was actually shot in HDR, and then check the YouTube app's quality settings - basically all of the resolution options should mention "HDR".