how to update current logged user username

Try hooking in as soon as the user is created or updated.

<?php

function wpse_250785_sync_user_login_with_email($user_id) {
    $user = new WP_User($user_id);

    if ($user->user_login != $user->user_email) {
        $user->user_login = $user->user_email;
        wp_update_user($user);
    }
}
add_action('user_register', 'wpse_250785_sync_user_login_with_email');
add_action('profile_update', 'wpse_250785_sync_user_login_with_email');

Disclaimer: Untested, but should work 😉