In fairness to Amazon they are normally very good at appeasing angry customers, usually with gift cards or if you are a Prime member by extending your Prime membership by 30 days free of charge.
However, sometimes if you are really annoyed about something you can push Amazon customer service to the point where their processes break down. Once you exhaust what they have a process for you will hit a brick wall and will eventually end up getting hung up on (regardless of how polite you may be) or unceremoniously kicked out of a live chat.
So, how do you raise a formal complaint with Amazon UK?
It's actually quite simple, just send an email to firstname.lastname@example.org
As a rule Amazon seem to want to make it as hard as possible to raise a formal complaint, it takes a lot of persuasion to reach the point where someone will hand out the information necessary to escalate above customer service. With this complaints email address however you should be able to resolve your complaint quite quickly. Even better the people on the end of this email appear to be located in the UK and not at an overseas contact centre like regular customer service.
Hopefully this helps you, let me know if you used this email address and whether they were able to resolve your issue once you complained. Good luck!