Issue with wp_update_user

I have some custom code where I allow an user to change his name, lastname and email through a form.

On form submission, I handle the _POST data to set the desired values for the current_user, but this seems to work only with custom metadata and update_user_meta(). These values do get changed and stay with the new setting.

Any call to wp_update_user() causes a weird “temporary” update, showing the updated values only the first time that the page handling _POST data is loaded. Any subsequent loading of that same page shows the original values. This means I change say the firstname from John to Peter and I see Peter only when I load the page for the first time. Then John comes back.

This is the code of the handler:

/* get current user and set up script vars */
$current_user = wp_get_current_user();
$id = $current_user->ID;
$key = '';
$value = '';
/* update custom meta field phonenumber */
if(isset($_POST['updated_phone'])) {
	$key = 'user_customfield_phonenumber';
	$value = $_POST['updated_phone'];
	update_user_meta($id,$key,$value);
}
/* update field firstname */
if(isset($_POST['updated_name'])) {
	$value = $_POST['updated_name'];
	wp_update_user( array ( 'ID' => $id, 'user_firstname' => $value ) ) ;
}

This is how I show the user’s details after calling the handler, the code is clean of all the formatting:

/* get current user and set up script vars */
$current_user = wp_get_current_user();
$id = $current_user->ID;
$tempkey = '';
$single = true;
/* display users firstname */
echo 'First Name: ' . $current_user->user_firstname;
/* display users custom meta phone */
$tempkey = 'user_customfield_phonenumber';
echo $current_user->__get($tempkey);

Any ideas?