How can I use the entry UID in the entry URI
I think this can be a starting solution. It rewrite slug of Entry after the Entry is saved to database. The URI is constructed from it. You can try it in any plugin, only register the EVENT_AFTER_SAVE_ELEMENT in init() function of Plugin.

use craft\events\ElementEvent;
use craft\services\Elements;

Event::on(
    Elements::class,
    Elements::EVENT_AFTER_SAVE_ELEMENT,
    function (ElementEvent $event){
        //identify the right Entry for example test sectionId, typeId, authorId
        if ($event-&gt;element instanceof craft\elements\Entry) {  
            $entry = $event-&gt;element;
            //only newly created 
            if($event-&gt;isNew){                  
                $entry-&gt;slug = $entry-&gt;uid;
                Craft::$app-&gt;elements-&gt;saveElement($entry);                 
            }
        }
    }
);