Basically UK consumer law means you have rights with the original seller, rather than the OEM. I'm stipulating that message indicates it was not purchased through Apple and instead through a reseller channel (more than likely EE if it's carrier locked).
If EE aren't willing to help then it might be a bit of an initial struggle to take it up with them if you weren't the original owner or still have the POP. However as a starting guide I'd suggest the following:
1) Take it to an Apple Store and see what the financial damage is for the repair.
2) Confirm if this is a warranty failure – although your sister babies the product, it may have been liquid damaged in the possession of the previous owner.
3) See if Apple can confirm who the original reseller was and they can hopefully advise the best course of action.
It's also worth reminding that consumer law is not an extended warranty so don't consider it as such.
If it's confirmed that the product failed of its own accord and you have the POP from the original reseller, this is certainly your best chance to get the product repaired or replaced under consumer law.
