Your right to a refund, in respect of faulty goods, only lasts until the point when you've legally 'accepted' them. The length of that period isn't defined in law. With many items it might only last a day or so, just giving you time to get the goods home and check that everything seems to be in order. However there was a case where a court ruled that a woman who had bought a car in the winter months hadn't legally accepted it at the point, in the summer, when she found out that the air conditioning wasn't working. (The court ruled that, because she'd had no reason to test the air conditioning before then, the 'pre-acceptance' appraisal period had not ended).
So, if you had bought your wellies in the summer sales, and not found out that they were defective until you first wore them in the winter, it's likely that you'd still have the right to a refund. However, since you actually reached a point (a few days after you first wore the wellies) when you were apparently satisfied with them, you've almost certainly 'accepted' the wellies. So your only right now is to a repair (which is obviously impractical) or a replacement.
The absence of a receipt doesn't affect your right to a replacement.
Chris