Here is the actual problem that occurred in your case.
When you embed a youtube video with the following code <iframe src="https://www.youtube.com/v/VIDEO_ID"></iframe>
a flash player is actually loaded. In this case your mobile doesn't support flash.
To rectify this issue you can use an HTML5 player which has the following structure <iframe src="https://www.youtube.com/embed/VIDEO_ID"></iframe>
You can use this youtube code generator for generating customized player embed code.