Did you pay by card? Then you'll have the transaction on your statement. If it's faulty they should take it back or at least swap it.
I bought 2 pairs of shoes from Bertie last year. The heels on both broke within days. I didn't have my receipts and had this big rant planned in my head for when I took them back. They put the money straight back on my card and I didn't even need to use my long angry tirade! I think if the product's faulty most people will do an exchange.
Like Ugly bob has already said- without proof of purchase you're unable to prove when this item was bought. Yes, if it's unfit for the purpose you are entitled to a refund HOWEVER this rule only applies if it's within a reasonable timescale that ya brolly broke, and regrettably you can't prove that.
However, there's no harm in seeing if Next will accept your word for when it was purchased- you never know :-)