Update (27/11/2020)

You can now use this method.

<?php
// Disable New User Registration email sent to Admin
add_action( 'init', function() {

	remove_action( 'register_new_user', 'wp_send_new_user_notifications' );
	remove_action( 'edit_user_created_user', 'wp_send_new_user_notifications', 10, 2 );
	
	add_action( 'register_new_user', 'bp_send_new_user_notifications' );
	add_action( 'edit_user_created_user', 'bp_send_new_user_notifications', 10, 2 );
	
} );

function bp_send_new_user_notifications( $user_id, $notify = 'user' ) {
	wp_new_user_notification( $user_id, null, $notify );
}
?>

Reference: https://wordpress.stackexchange.com/questions/236122/turn-off-new-user-registration-emails/236301#236301

Original Post

When a new user is registered, WordPress sends a New User Registration notification to the admin email address in the site’s General Settings. A Your username and password email is also sent to the new user.

This functionality is handled by wp_new_user_notification() located in wp-includes/pluggable.php.

<?php
if ( !function_exists('wp_new_user_notification') ) :
/**
* Notify the blog admin of a new user, normally via email.
*
* @since 2.0
*
* @param int $user_id User ID
* @param string $plaintext_pass Optional. The user's plaintext password
*/
function wp_new_user_notification($user_id, $plaintext_pass = '') {
    $user = get_userdata( $user_id );

    // The blogname option is escaped with esc_html on the way into the database in sanitize_option
    // we want to reverse this for the plain text arena of emails.
    $blogname = wp_specialchars_decode(get_option('blogname'), ENT_QUOTES);

    $message = sprintf(__('New user registration on your site %s:'), $blogname) . "\r\n\r\n";
    $message .= sprintf(__('Username: %s'), $user->user_login) . "\r\n\r\n";
    $message .= sprintf(__('E-mail: %s'), $user->user_email) . "\r\n";

    @wp_mail(get_option('admin_email'), sprintf(__('[%s] New User Registration'), $blogname), $message);

    if ( empty($plaintext_pass) )
        return;

    $message = sprintf(__('Username: %s'), $user->user_login) . "\r\n";
    $message .= sprintf(__('Password: %s'), $plaintext_pass) . "\r\n";
    $message .= wp_login_url() . "\r\n";

    wp_mail($user->user_email, sprintf(__('[%s] Your username and password'), $blogname), $message);

}
endif;
?>

This is a pluggable function so it can be customized by redefining it . Unfortunately redefining it in your theme has no effect, it needs to be replaced via a plugin.

It seems that plugins are loaded first, then pluggable.php, then the theme. So if your function override is in your theme, you’ll clash with the default function. – Steve Taylor

Here’s a quick plugin template for customizing the notifications.

<?php
/**
 * Plugin Name: Custom New User Notifications
 * Description: Customize the "New User Registration" and "Your username and password" notifications
 * Version: 1.0
 * Author: Ben Pearson
 * Author URI: http://benpearson.com.au
 */

if ( !function_exists('wp_new_user_notification') ) :
/**
 * Notify the blog admin of a new user, normally via email.
 *
 * @since 2.0
 *
 * @param int $user_id User ID
 * @param string $plaintext_pass Optional. The user's plaintext password
 */
function wp_new_user_notification($user_id, $plaintext_pass = '') {
    $user = get_userdata( $user_id );

    // The blogname option is escaped with esc_html on the way into the database in sanitize_option
    // we want to reverse this for the plain text arena of emails.
    $blogname = wp_specialchars_decode(get_option('blogname'), ENT_QUOTES);

    $message  = sprintf(__('New user registration on your site %s:'), $blogname) . "\r\n\r\n";
    $message .= sprintf(__('Username: %s'), $user->user_login) . "\r\n\r\n";
    $message .= sprintf(__('E-mail: %s'), $user->user_email) . "\r\n";

    @wp_mail(get_option('admin_email'), sprintf(__('[%s] New User Registration'), $blogname), $message);

    if ( empty($plaintext_pass) )
        return;

    $message  = sprintf(__('Username: %s'), $user->user_login) . "\r\n";
    $message .= sprintf(__('Password: %s'), $plaintext_pass) . "\r\n";
    $message .= wp_login_url() . "\r\n";

    wp_mail($user->user_email, sprintf(__('[%s] Your username and password'), $blogname), $message);

}
endif;
?>

Include the function_exists() check it’s wrapped in. If you include this, your function will still get priority over the WP core; it’ll just prevent fatal errors if you activate a plugin that happens to have overridden the same function. – Steve Taylor

Resources

• wp_new_user_notification() in the Codex
• Customizing the new user email with a pluggable function by Steve Taylor