Fixed it!

In the previous code the only issue was the video wasn't loading on post's sidebar, as the video was hidden initially with inline css and the script which was supposed to make the video visible in case of no error, was not running in post's sidebar.

New code -

<video width="100%" height="auto" poster="https://webtik.in/ads/cover.jpg?nocache=<?php echo time(); ?>" controls style="display: block;" onerror="this.style.display='none';">
    <source src="https://webtik.in/ads/video.mp4?nocache=<?php echo time(); ?>" type="video/mp4" onerror="this.parentElement.style.display='none';">
</video>

Updating the question from "Make javascript/jQuery script run after the sidebar loads dynamically" to "Bulk upload a video on multiple WordPress websites at once by just uploading the video on a server" in case anyone wants to achieve similar thing. Cheers!