Custom Post Type Template Insert not working properply

Short answer: I guess the problem of your code is the missing die(); as last line of doThemeRedirect method.

After that, your code can be simplified a lot, using the template_include filter hook instead of the template_redirect action hook.

Simplify means reduce the chances of bugs, or at least, let recognization of them simpler.

 public function __construct() {
    add_filter( 'template_include', array( $this, 'uniThemeRedirect') );
}


public function uniThemeRedirect( $template ){
    if(  get_query_var('post_type') == 'videos' || (is_single() && get_post_type() == 'videos') ) {

        // the 'single-videos.php' will be included for both
        // single and archive view of CPT 'videos'
        // to stop including for single view, remove
        // (is_single() && get_post_type() == 'videos') ) from if condition
        // to stop including for archive view, remove
        // get_query_var('post_type') == 'videos' from if condition

        $templateFilename="single-videos.php";
        $locate = locate_template( $templateFilename );
        if( $locate ) return $locate;
        if( file_exists( user_trailingslashit(UNISLIDER_TEMPLATES) . $templateFilename ) )
           return user_trailingslashit(UNISLIDER_TEMPLATES) . $templateFilename );
    }
    // die 'not working';
    return $template;

}

So you don’t need the doThemeRedirect method at all.

This code is simpler than your, but more powerful (it supports child themes where the yours doesn’t) and a bit more good-practised: there is no one global variable in it.