Login redirect to home page doesn’t update nav

I am using the following code to redirect the user to the home page after they login, but the primvary nav doesn’t update to the logged in users nav after the redirect; I am logged in technically because if I refresh or go to another page the correct nav loads. Any idea how to fix that?

add_filter('woocommerce_login_redirect', 'wc_login_redirect');
function wc_login_redirect( $redirect_to ) {
     $redirect_to = home_url();
     return $redirect_to;
}