That was easier than I expected!

I've updated my comments.php so that replies are now ordered chronologically and you naturally read the conversation naturally down the page.

It now uses a WP_Comment_Query to pull the relevant comments from the database where the webmention type, as detected by the Semantic Linkbacks plugin, is 'reply' and I just run through them with a foreach loop.
 

$args = array(
  'post_id' => $id,
  'meta_key' => 'semantic_linkbacks_type',
  'meta_value' => 'reply',
  'order' => 'ASC'
);

$replies_query = new WP_Comment_Query;
$replies = $replies_query->query( $args );