What template files do I need to customise custom-category-term-links rather than fall back on archive.php?

Custom taxonomies have templates named taxonomy-something in this hierarchy:

taxonomy-{taxonomy}-{term}.php
taxonomy-{taxonomy}.php
taxonomy.php
archive.php
index.php

So, yes, if you have no taxonomy-custom-cat-term-cat.php, taxonomy-custom-cat.php or taxonomy.php then archive.php will be used.

category-something.php will only be used for the built-in categories, if you are on a page showing posts from a category called “something.”