I’ve narrowed down the issue a bit more. The page is loading the billing template (templates/checkout/form-billing.php), since it shows any HTML I put in there. However, the following lines seem to be giving no output at all:
<?php foreach ($checkout->checkout_fields[‘billing’] as $key => $field) : ?>
<?php woocommerce_form_field( $key, $field, $checkout->get_value( $key ) ); ?>
<?php endforeach; ?>
Hope anyone knows where to look from here!