I have a copy of the date from my end when transfer was done & notification via email when they had received payment if that helps as well as the invoice.
I'll look into trading standards. Difficult to say anything to them directly as they won't answer my calls.
Trading Standards won't get you a refund, or tackle your specific case for you. But by letting them know, and if other people complain about this seller as well, then they can choose to take action against them.
It's tricky to get a refund where there's no specific mechanism in place (such as ebay's guarantee) and the seller refuses to answer. Try Citizens Advice if you have no reply from them at all. But covid has made all this much more difficult.
I don't know how much your item cost, and I know this doesn't help you in this instance, but if you are buying anything that costs more than £100 (i.e £100.01p or more) then
always pay by credit card, as under Section 75 of the CCA the credit card company becomes jointly liable to you for the correct fulfilmnt of the transaction, and will refund you if the original seller is mucking you about. Even if you pay just the deposit or even just 1p of the total cost on credit card they are liable for the full amount.
Good luck