Get WordPress to use another custom “author.php” file (“custom-author-file.php”)

In author.php:

<?php
get_template_part('custom-author-file');

Alternatively, rename your custom-author-file.php to author.php.

The URL being used in this case isn’t relevant, as a template doesn’t determine the URL used, it’s the other way around. However, I suspect there is more to your question you haven’t explained