If the date in question is all on one line (ie not being shown in a special layout), you can edit the template/widget and locate the date tag, which looks something like this:
<?php echo date('d M Y', strtotime($post->post_date)); ?>
And replace it with a generic tag that will take the format you set in Settings > General:
<?php the_time(get_option('date_format')); ?>
If the date is shown in a special layout, such as the Handmade Two post date flags or the Ambition Dater blocks, you must edit the format within the existing tag to avoid breaking the layout.
Example of Handmade Two’s date tag in fetch-list.php. M and jS are format codes corresponding to the English month and day.
<div>
<span><?php echo date_i18n('M', strtotime($post->post_date)); ?></span>
<span><?php echo date_i18n('jS', strtotime($post->post_date)); ?></span>
</div>
Example of Ambition’s Date Block in the fetch-list.php:
<h5>
<span><?php the_time('j') ?></span>
<span><?php the_time('M') ?></span>
<span><?php the_time('Y') ?></span>
</h5>
Refer to this post in the WordPress Codex for format code help and be very careful not to remove the single quotes when editing these values: