Pagination for normal (standard) posts on a page with a custom loop?

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get_pagination_links is looking at $wp_query, whereas you want pagination for your query object named $loop. So instead, use paginate_links

Something similar to this code, which is based on get_pagination_links, alebit with wp-query swapped out for your loop query object:

$endsize = $midsize = 1;
$type="plain";
$loop->query_vars['paged'] > 1 ? $current = $loop->query_vars['paged'] : $current = 1;

// Sanitize input argument values
if ( ! in_array( $type, array( 'plain', 'list', 'array' ) ) ) $type="plain";
$endsize = (int) $endsize;
$midsize = (int) $midsize;

// Setup argument array for paginate_links()
$pagination = array(
    'base'          => @add_query_arg('paged','%#%'),
    'total'         => $loop->max_num_pages,
    'current'       => $current,
    'show_all'      => false,
    'end_size'      => $endsize,
    'mid_size'      => $midsize,
    'type'          => $type,
    'prev_next'     => false,
    'paged'         => $loop->query_vars['paged']
);

echo paginate_links( $pagination );

Finally you’ll need to grab the paged URL variable via $_GET, do a tad sanitisation ( check it’s a number and it’s set ), then put it into your WP_Query args array