Just adding to the answers by Thomas Griffin and user114196 (not got the reputation to comment). The hook profile_update is also called when the lost password functionality is used, before the email is sent, so the redirect breaks this in a very annoying to debug way. This can be fixed by checking if user_activation_key has changed (the link used for password reset), and not redirecting if so.
add_action('profile_update', 'wbb_profile_update', 10, 3);
function wbb_profile_update($user_id, $oldUserData, $newUserData) {
// This is also called on lost password, BEFORE the email is sent - hence breaking it. If that's the case, don't redirect.
$doRedirect = $oldUserData->data->user_activation_key === $newUserData['user_activation_key'];
if ($doRedirect && get_option('wbb-redirect-users-to-on-login', '')){
wp_redirect(get_option('wbb-redirect-users-to-on-login', ''));
die();
}
}