no-notifications.php

<?php
/**
 * Plugin Name: Turn off new user registration emails
 * Description: Turns off new user registration emails just for the admin. No other notifications are affected.
 * Plugin URI:  http://torstenlandsiedel.de
 * Version:     1.0
 * Author:      Torsten Landsiedel
 * Author URI:  http://torstenlandsiedel.de
 * Licence:     GPL 2
 * License URI: http://opensource.org/licenses/GPL-2.0
 */

if ( ! defined( 'ABSPATH' ) ) {
	exit; // Exit if accessed directly.
}

/*
 * See: https://wordpress.stackexchange.com/questions/236122/turn-off-new-user-registration-emails
 */
add_action( 'init', function() {
	remove_action( 'register_new_user', 'wp_send_new_user_notifications' );
	add_action( 'register_new_user', 'wpse236122_send_new_user_notifications' );
} );

/**
 * Initiates email notifications related to the creation of new users.
 *
 * Notifications are sent just to the newly created user.
 *
 * @since 4.4.0
 * @since 4.6.0 Converted the `$notify` parameter to accept 'user' for sending
 *              notifications only to the user created.
 *
 * @param int    $user_id ID of the newly created user.
 * @param string $notify  Optional. Type of notification that should happen. Accepts 'admin'
 *                        or an empty string (admin only), 'user', or 'both' (admin and user).
 *                        Default 'both'.
 */
function wpse236122_send_new_user_notifications( $user_id, $notify = 'user' ) {
	wp_send_new_user_notifications( $user_id, $notify );
}

/**
 * Disable call in BuddyPress.
 *
 * @link https://github.com/buddypress/BuddyPress/blob/fe47890c1ab7e2cb24c005f1c35fb9c6c8c8ab7c/src/bp-members/bp-members-functions.php#L1891-L1900
 */
add_filter( 'bp_core_send_user_registration_admin_notification', '__return_false' );