How to insert date in topNav

wp_nav_menu has a $container parameter that can be used to set the type of container (div or nav) or to disable it.

You could set the parameter to false and hardcode the wrapper div whilst including a date inside of it:

<!-- BEGIN HEADER-TOP -->
<div class="navigation-wrap">
    <?php wp_nav_menu( array(
        'theme_location' => 'topNav',
        'depth' => 3,
        'fallback_cb' => false,
        'menu_class' => 'topNav',
        'container' => false
    )); ?>
    <div class="navigation-date">
        <?php echo date('l, jS \of F Y'); ?>
    </div>
</div>
<!-- END HEADER-TOP -->