Check out Caxton cards. Someone else recommended this card to me, as a PAYG when going on holiday. I have not looked into it at all.
Although you are applying for a PAYG card, as a result of most card companies now withdrawing from PAYG Credit Cards, for MLA purposes, etc., you will still be asked to provide personal details, and most probably you will be asked to set up the account online, which will also require you to give your personal details.
I'd be very interested to see if you are successful in getting an over the counter credit card with no ID requirments. Even if you do, any top-ups over a certain minimum account will then require you to register in order to continue to use the card.