Create an user on external database

No matter how you create a new user, it will end up calling wp_insert_user, which runs the action user_register right before finishing. So you can do something like the following:

add_action('user_register','register_user_remotely');

function register_user_remotely($user_id) {
   //add user to external database
   $url = ...; //url of file that will receive the $user->data in the $_POST variable
   $user = get_user_by('id', $user_id);
   if (!is_wp_error($user)) {
      $response = wp_remote_post($url,array('body' => $user->data));
      if (is_wp_error($response)) {
         ...
      } else { ... }
   }
}

This will post all user data from the wp_users table to the project management app. As pointed out by user42826, you’ll then have to use its API to create the user at that end.

$user->data['user_pass'] contains the user’s encrypted password, but this is useless unless the app encrypts its password just as WordPress. Seeing the front end of your app, probably the best option is to create an arbitrary password in the app and make it send a mail to the user with a password reset link.