Hi,
Unite Revolution Slider is not developed by us. It is a 3rd party plugin that was injected to our template to make things look better and nicer. Probably YouTube has blocked some of API's that Unite Revolution Slider was using. You should better upgrade the Unite Revolution Slider from its original source to avoid this YouTube API problem:
www.unitecms.net/joomla-extensions/unite-revolution-slider-responsive
Alternatively, download the video from YouTube
(use google to find a way to download videos from YouTube), upload that video to your own website and insert it to slider as HTML
Regards.