Hard Drive Fail

The key code for Windows 8 is embedded into the system board - as long as you use a Dell OEM Windows media (disc or flash drive) you don't need a product key.

I don't have a Dell OEM disk as I wasn't given one, and the USB backup I made isn't working. Is there a way around this?

The USB flash drive requires that secure boot be turned off before it'll boot to restore the system.  If that doesn't work, you will need to contact Dell for the recovery media (they'll generally ship it free of charge under warranty, or for about $50 if your warranty has expired).

I tried to do a recovery with the secure boot off but it still did not install. 

I live in the UK and unfortunately Dell in the UK do not provide you with recovery discs. Isn't there anyway I could find out what my product key was?

It won't matter - the key will work only with a Dell OEM Windows recovery media.  You'll need to contact Dell UK for that (in warranty, no charge; out of warranty, there is a charge).

I live in the UK and unfortunately Dell in the UK do not provide you with recovery discs. Isn't there anyway I could find out what my product key was?

Sure there is, see here:

http://dellwindowsreinstallationguide.com/a-clean-install-of-windows/microsoft-product-activation/obtaining-your-oem-bios-embedded-slp-product-key-using-rw-everything-oem-only/

However the key will not help you unless you have access to a Windows 8.1 retail .iso.

If you think you can use your OEM Windows product key with the Microsoft installer forget here:

http://dellwindowsreinstallationguide.com/download-microsoft-windows-and-office/download-microsoft-windows/download-windows-8-1-iso/