My first thought is that this would probably work if it weren’t a multiselect. It being a multiselect means that $value will be outputting an array of users. You’ll need to stick a foreach loop in there to confirm, something like:
<div class="module_col">
<h3><?php the_sub_field('module_title'); ?></h3>
<?php
// Get List of students to check for eligibility
$students_list = get_sub_field('student_availability' );
// Set the default state to false, to be changed later
$student_authorised = false;
// Loop through students_list to check if user is allowed access
foreach( $students_list as $student ) :
// If user is in the list, set student_authorised to true and break out of loop
//(no need to loop through everybody if they're the first user)
if( in_array( $student_username, $student['nickname'] ) )
$student_authorised = true;
break;
endforeach;
// If student is in the list, show download link
if ( $student_authorised ) : ?>
<a href="<?php the_sub_field('module_download'); ?>">
<button class="download_btn">
<span><i class="fa fa-cloud-download"></i>Download Module</span>
</button>
</a>
<?php
// if student not in list, show authentication error
else : echo 'Not logged in'; endif; ?>
</div>
