I have found what is causing the issue in my case.
This widget, en-queues the twitter script using a protocol relative url. And I am using bwp-minify plugin, which apparently doesn't support protocol relative urls, so it doesn't include widget.js script!
As a fix, I have added the following line to my functions.php which registers the script with an absolute url:
wp_register_script( 'twitter-widgets', 'http://platform.twitter.com/widgets.js', '', '', true );