Thanks for your report, Bradley!
Taking a look at Fanfiction.net, we found that the stories have two links, one to the first chapter of the story and the other as a quick link to the latest chapter of the story. The second link is, of course, only present when there is more than one chapter for the story.
Without F123 Access, the quick link to the latest chapter won't be labeled. NVDA just simply reads it as 'Link'. With F123 Access, the automated link labeling scripts label this quick link properly, which is why you found the link to the last chapter.
The usual link to the first chapter of the story is still there, before the quick link. If this is not the case for you, and you can only see the quick link, please submit a report and we will get to fixing the issue.