YouTube is beginning to send out an update to users which flips the switch on a set of minor UI changes, most notably of those being the repositioning of the navigation within the app, which users will now be able to interact with via the bar that has a bottom placement under all of the content that's being displayed on screen. This is a server side change, which means that users won't have an update to look for in Google Play and instead will simply have to wait until it shows up. That also means there is really no time frame for this as Google didn't announce the changes.
Having said that, you can tell if you have the new UI by opening up YouTube and if the buttons for switching between the home, trending, subscription, shared, and account tabs are all still sitting at the top in the header bar, then the UI changes have not yet taken affect. What you should see is a white bar at the bottom of the app UI which has all of those buttons. This is also not the only change that will come along with this server side update. By moving the navigation to the bottom of the app's UI Google was able to free up some space up top, and this has resulted in a smaller header.
Other than the smaller header bar up top and the relocated navigation buttons, which have also had their colors swapped from white to red to account for the difference in color of the bar itself, there doesn't appear to be any other changes to watch for here. So, these are indeed minor UI tweaks, but they seem less minor when you consider them in pairing with other UI changes that have already been implemented in the past recently. Although the changes haven't yet hit everyone, the YouTube app on Android also recently started appearing for some users with a new comments section UI, and just back at the end of August some users started noticing a new video description UI, although it is worth mentioning that not all uses are likely seeing this visual change either.