Tesco normally respond within a week or so to postal complaints. So it's certainly worth chasing up your original problem.
Some new ink cartridges (even those from the manufacturer) seem not to work until you've performed several 'deep cleans'. Assuming that you've done that (and, of course, that you bought the correct cartridge for your printer in the first place), Tesco must remedy your problem. They're definitely obliged to offer a replacement but it's also likely (because you've never legally 'accepted' the goods as satisfactory) that you can demand a full refund instead. The absence of a receipt is irrelevant; your rights remain unaffected. You simply need to show that you purchased the product at Tesco. Their brand name on the packaging is sufficient proof of that.
Chris