What to do with UUID of group of files?

In many cases you don't even need individual files UUIDs . You can address each file in a group by indexing the group:

http://www.ucarecdn.com/d51d18b3-5150-4153-b4c6-0a70f36663db~2/nth/0/
http://www.ucarecdn.com/d51d18b3-5150-4153-b4c6-0a70f36663db~2/nth/1/

See more here: https://uploadcare.com/documentation/cdn/#group-api

If you  really need files UUIDs, it is better to subscribe to onChange event using JavaScript API:

widget.onChange(function(group) {
  if (!group) {
    return;
  }
  group; // group object
  group.files(); // array of file objects
  $.when.apply(null, group.files()).then(function() {
    arguments; // array of individual files infos
  });
});