Как изменить письмо при смене пароля из личного кабинета

Для этого, нужно создать плагин в котором определить следующие функции. Они отвечают за то как письма будут выглядеть. Все такие функции находятся в файле wp-includes/pluggable.php

Вот нужные вам функции:

/**
 * Notify the blog admin of a user changing password, normally via email.
 *
 * @since 2.7.0
 *
 * @param WP_User $user User object.
 */
function wp_password_change_notification( $user ) {
	// send a copy of password change notification to the admin
	// but check to see if it's the admin whose password we're changing, and skip this
	if ( 0 !== strcasecmp( $user->user_email, get_option( 'admin_email' ) ) ) {
		$message = sprintf(__('Password Lost and Changed for user: %s'), $user->user_login) . "\r\n";
		// 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);
		wp_mail(get_option('admin_email'), sprintf(__('[%s] Password Lost/Changed'), $blogname), $message);
	}
}

/**
 * Email login credentials to a newly-registered user.
 *
 * A new user registration notification is also sent to admin email.
 *
 * @since 2.0.0
 * @since 4.3.0 The `$plaintext_pass` parameter was changed to `$notify`.
 * @since 4.3.1 The `$plaintext_pass` parameter was deprecated. `$notify` added as a third parameter.
 *
 * @global wpdb         $wpdb      WordPress database object for queries.
 * @global PasswordHash $wp_hasher Portable PHP password hashing framework instance.
 *
 * @param int    $user_id    User ID.
 * @param null   $deprecated Not used (argument deprecated).
 * @param string $notify     Optional. Type of notification that should happen. Accepts 'admin' or an empty
 *                           string (admin only), or 'both' (admin and user). Default empty.
 */
function wp_new_user_notification( $user_id, $deprecated = null, $notify = '' ) {
	if ( $deprecated !== null ) {
		_deprecated_argument( __FUNCTION__, '4.3.1' );
	}

	global $wpdb, $wp_hasher;
	$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(__('Email: %s'), $user->user_email) . "\r\n";

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

	// `$deprecated was pre-4.3 `$plaintext_pass`. An empty `$plaintext_pass` didn't sent a user notifcation.
	if ( 'admin' === $notify || ( empty( $deprecated ) && empty( $notify ) ) ) {
		return;
	}

	// Generate something random for a password reset key.
	$key = wp_generate_password( 20, false );

	/** This action is documented in wp-login.php */
	do_action( 'retrieve_password_key', $user->user_login, $key );

	// Now insert the key, hashed, into the DB.
	if ( empty( $wp_hasher ) ) {
		require_once ABSPATH . WPINC . '/class-phpass.php';
		$wp_hasher = new PasswordHash( 8, true );
	}
	$hashed = time() . ':' . $wp_hasher->HashPassword( $key );
	$wpdb->update( $wpdb->users, array( 'user_activation_key' => $hashed ), array( 'user_login' => $user->user_login ) );

	$message = sprintf(__('Username: %s'), $user->user_login) . "\r\n\r\n";
	$message .= __('To set your password, visit the following address:') . "\r\n\r\n";
	$message .= '<' . network_site_url("wp-login.php?action=rp&key=$key&login=" . rawurlencode($user->user_login), 'login') . ">\r\n\r\n";

	$message .= wp_login_url() . "\r\n";

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

Также, есть такие фильтры, которые срабатывают во время вызова функции wp_update_user():

Изменим текст письма при смене пароля:

add_filter( 'password_change_email', 'change_password_mail_message', 10, 3 );
function change_password_mail_message( $pass_change_mail, $user, $userdata ){
	$new_message_txt = __( 'Hi ###USERNAME###,

This notice confirms that your email was changed on ###SITENAME###.

If you did not change your email, please contact the Site Administrator at
###ADMIN_EMAIL###

This email has been sent to ###EMAIL###

Regards,
All at ###SITENAME###
###SITEURL###' );

	$pass_change_mail[ 'message' ] = $new_message_txt;

	return $pass_change_mail;
}

Изменим текст письма при смене почты:

add_filter( 'email_change_email', 'email_change_email_message', 10, 3 );
function email_change_email_message( $email_change_email, $user, $userdata ){
	$new_message_txt = __( 'Hi ###USERNAME###,

This notice confirms that your email was changed on ###SITENAME###.

If you did not change your email, please contact the Site Administrator at
###ADMIN_EMAIL###

This email has been sent to ###EMAIL###

Regards,
All at ###SITENAME###
###SITEURL###' );

	$email_change_email[ 'message' ] = $new_message_txt;

	return $email_change_email;
}