1) I have never applied for credit from Littlewoods.
2) The address on the letter was for the flat directly upstairs.
The incorrect address appears to be the reason I was turned down, as they couldn't link my name and that address.
So I emailed the police*, who told me that I could get a credit report and report any mistakes on it to the credit agency so they could be fixed. Not quite the response I was looking for, so I asked if nothing was going to be done about the people doing this. And their email said to phone up for a proper chat.
So I did. And was told that basically I have no direct bearing on this case. Someone is attempting to defraud Littlewoods, and failing. Which means that Littlewoods might have a case against them, but I certainly don't, as I'm only tangentially involved. The nice man at the other end of the phone _did_ tell me that I could sign up for a month free with the Equifax/Experian credit agency by going to creditexpert.co.uk/, and check my credit report that way.
Which shows that my credit report is completely clean, my credit score is 997 out of a possible 1000, and that I have nothing to worry about. So I emailed their fraud people anyway, just so I have something for my records. And I guess that's where I have to stop, unless anyone has any suggestions for anything else I can do**?
*They handily have an email address for reporting non-urgent crimes. Their contact page also states that they cannot accept reports for crimes over Twitter. I'm somewhat boggled that people would try.
**Short of drilling through the floorboards into the upstairs flat and releasing lighter than air poison gas.
Original post on Dreamwidth - there are comments there.