Please watch the demo at: https://www.wonderplugin.com/wordpress-gallery/continuous-youtube-playing-demo/
The demo is created with WonderPlugin Gallery Plugin.
To play YouTube video continuously, in the plugin, Step 3, Options, tab, check the option "Auto slide" and "Automatically play video".
Please note, this feature won't work on mobile and tablet devices, for example iPhone, iPad and Android. It also won't work on old web browsers, for example, Internet Explorer 8.