At present, the video shortcode does not support mobile devices due to a numbers of reasons – restrictions on videos on mobile devices due to battery concerns and limited resources, flash support. Best way to go about this is to have two separate sections for mobile and desktop and then use media queries or CSS to hide/show the appropriate sections on the respective devices. So, you can have two page sections desktop-video and another mobile-video. For mobile video, you can use the regular youtube video URL. Then use CSS like this
.mobile-video {display: none;}
.mobile-device .desktop-video {display: none;}
.mobile-device .mobile-video {display: block;}