Subpages getting a 404 error after changing Parent page URL

Based on what you wrote above I do not see the problem of changing the page name would result in 404 errors on sub-level pages.

What you should look for

1- Check if you do not have active plugins like (The Event Calendar) or equivalents. Those plugins are already using, most of the time, those slug or rewrite rules.

2- You did not mentionned that you were using Custom Post Type, but if it would be the case, you can edit the rewrite rule to place your new slug (Codex : register_post_type)

function register_custom_post_type_example() {
    $rewrite = array(
        'slug'                  => 'calendar',
        'with_front'            => true,
        'pages'                 => true,
        'feeds'                 => true,
    );
    $args = array(
        'label'                 => 'Your label'
        'description'           => 'Your description',
        'supports'              => array('page-attributes'),
        'taxonomies'            => array(),
        'hierarchical'          => false,
        'public'                => true,
        'show_ui'               => true,
        'show_in_menu'          => true,
        'menu_position'         => 20,
        'menu_icon'             => 'dashicons-networking',
        'show_in_admin_bar'     => true,
        'show_in_nav_menus'     => true,
        'can_export'            => true,
        'has_archive'           => true,
        'rewrite'               => $rewrite,
    );
    register_post_type( 'equipe', $args );
}
add_action( 'init', 'register_custom_post_type_example' );

In the code above, you can see that you can change the rewrite rule. But as I said, do not try to work with alot of plugin because it will results in pages conflicts.

Hope it can guide you to find the problem.