First time trying prestashop (at a customer place) on an old version (1.5.3). I'm just trying to figure out how to get the delivery address details in order-payment.tpl
[EDIT] Ok, found out by myself, you have a "super" variable in smarty who contains all you need : $delivery, so $delivery->firstname, $delivery->lastname and so on ...
I using a great module named "Delivery Automation QR-code Module" on PrestaShop Addons which which automate and simplify my delivery system using QR-codes and advise you to try it.