and was told the computer had been written off and she was to go to the shop with the supplied referance number to collect a free replacement."
That's the part to concentrate on. Your cousin was supplied with a reference code for a replacement computer. Presumably PC World hasn't denied that this code was issued, or that it isn't a valid one? Your cousin accepted the offer, but a PC World branch has refused to honour the agreement.
Please tell your cousin to write to:-
Customer Services Department
DSG Retail Ltd
Telling them what happened in a clear and concise way. She should tell them that she has asked us for advice, and that I have suggested she write to them as she has received no satisfaction from the branch. Tell them that we'll be monitoring this case in our forum.
Under the terms of current consumer law your cousin is entitled to a repair or replacement if a fault develops within six months of the date of purchase. The law is perfectly clear that the supplier is liable in a case like this.
The letter should be sent by Special delivery - not recorded delivery.
Please update us as soon as your cousin hears anything in response, or if she receives no reply within 7 days of the date of her letter.