Send An Email to Admin on User Profile Completion

Based on your question, you could simply call the WordPress function

wp_mail( $to, $subject, $message, $headers );

when your var $user_progress['completion_percentage'] equal 100 %.

<?php 
if( $user_progress['completion_percentage'] === 100 ) {
  // Complete your variables with original code
  $to = "[email protected]";
  $subject = __("New user registration", "yourdomain");
  $headers = array('Content-Type: text/html; charset=UTF-8');
  $message = "A new user have completed his/her profile" . $user['id'];
  
  // Send the email
  wp_mail( $to, $subject, $message, $headers );

}
?>

Source : https://developer.wordpress.org/reference/functions/wp_mail/