The problem with giving a legal opinion is that much of English law is defined by precedents (i.e. court decisions) rather than by anything that was actually put into writing by Parliament. Unless a court has ruled upon whether a receipt is required
and on the form that the receipt should be in (and in a way that covers all types of purchases) it's probable that nobody knows exactly what the law is, simply because the law hasn't yet been fully defined.
The Government website states:
"You do not need a receipt to obtain a refund for faulty goods. However, you may be required to show proof of purchase with a credit card slip or bank or credit card statement"
https://www.gov.uk/government/news/how-to-be-savvy-when-doing-your-online-shopping
(Although the title of that page refers to online shopping, the actual content refers to buying on the High Street).
However the actual wording used on other 'authoritative' websites varies and it's likely that any evidence that the item came from the store to which it's being returned (such as the fact that no other retailer sells that brand or the store's price ticket still being attached) would suffice. It's also possible that a County Court would uphold a simple statement from the purchaser (with no documentation whatsoever), as to where he/she bought the item, based solely upon the 'balance of probabilities'.
I've had a full refund from Maplin for a power supply that failed within a few weeks of purchase, even though I had no receipt. (I had to hassle a bit but I did point out that it was their own brand, so it must have come from them). Similarly I've had a full refund from Sainsbury's for a laminator that packed up shortly after purchase (again without a receipt) and Tesco offered me a full refund (although I accepted a replacement anyway) when I returned a faulty modem without a receipt.