When you run your foreach loop, you can also define an index variable. And you can use that index variable to get the prev and next.
Something like:
<?php
$project_gallery_thumbnails_images = get_field( 'project_gallery_thumbnails' );
// let store a total count for later use
$project_gallery_thumbnails_count = count($project_gallery_thumbnails_images);
?>
<?php if ( $project_gallery_thumbnails_images ) : ?>
<?php foreach ( $project_gallery_thumbnails_images as $index => $project_gallery_thumbnails_image ): ?>
<a class="thumbnail-link" href="#image_<?php echo $project_gallery_thumbnails_image['id']; ?>">
<img src="<?php echo $project_gallery_thumbnails_image['sizes']['thumbnail']; ?>" alt="<?php echo $project_gallery_thumbnails_image['alt']; ?>" />
</a>
<div id="image_<?php echo $project_gallery_thumbnails_image['id']; ?>" class="lb-overlay" >
<?php if ($index != 0): // dont do it on the first one ?>
<a href="#image_<?php echo $project_gallery_thumbnails_images[$index - 1]['id']; ?>">Previous Gallery Image ID</a>
<?php endif; ?>
<a class="thumbnail-link-close" href="#page">
<img class="lazyload" src="<?php echo $project_gallery_thumbnails_image['url']; ?>" alt="<?php echo $project_gallery_thumbnails_image['alt']; ?>" />
</a>
<?php if ($index != $project_gallery_thumbnails_count - 1): // dont do it on the last one ?>
<a href="#image_<?php echo $project_gallery_thumbnails_images[$index + 1]['id']; ?>">Next Gallery Image ID</a>
<?php endif; ?>
</div> <!--lb-overlay-->
<?php endforeach; ?>
<?php endif; ?>
Cheers.
