How to add current custom taxonomy slug to body class

You can follow similar approach with get_the_terms.
Here is the function.

function add_taxonomy_to_single( $classes ) {
    if ( is_single() ) {
        global $post;
        $my_terms = get_the_terms( $post->ID, 'custom-taxonomy' );
        if ( $my_terms && ! is_wp_error( $my_terms ) ) {
            foreach ($my_terms as $term) {
                $classes[] = $term->slug;
            }
        }
        return $classes;
    }
}
add_filter( 'body_class', 'add_taxonomy_to_single' );

Don’t forget to change the name of your custom taxonomy in above code. I used custom-taxonomy for example.

EDIT 1:

To add first taxonomy name in body class here it the updated code.

function add_taxonomy_to_single( $classes ) {
    if ( is_single() ) {
        global $post;
        $my_terms = get_the_terms( $post->ID, 'custom-taxonomy' );
        if ( $my_terms && ! is_wp_error( $my_terms ) ) {
            $classes[] = $my_terms[0]->slug;
        }
        return $classes;
    }
}
add_filter( 'body_class', 'add_taxonomy_to_single' );