I have never worked with video libraries in PHP, but I can tell you exactly what you need to do:
First, you need to create a PHP file that does not have (or has a very large) time limit for work, and also, which does not stop its work if the user disconnects the connection. This can be implemented by a number of ini_set () t functions. In this file to connect the library work with video files. The script will first have to check if the file is broken. Next, convert the resulting video to mp4, and then make several versions with different quality (140p, 240p, 360p, 480p, 720p, 1080p, ...) If the video file is large (several gigabytes) it can take up to several hours.
I do not know why you needed to create such a service, because there are already such services as YouTube, RuTube, VK, which have all these functions, why reinvent the wheel, which has already been invented many times already.
On the question of how VK notifies the user about the end of processing. If you are interested in the notification process itself, then VC sends Ajax - requests for a specific script every 30 seconds, which returns JSON - an array with new notifications, if any. Also received and messages, add to friends, and other notifications. Perhaps I misunderstood your question. In this case - do not beat with sticks.