Lounge access question, grateful for any advice.

Booked economy ticket EDI-IST-RUH. One way.

I’ve no status with TK or * alliance.

Post-purchase, I took the upgrade offer for the EDI-IST leg, paid with cash (not miles). TK website says no lounge access is included these offers if paying cash. However, at EDI check-in I was handed an access card for the contract lounge they use there.

On arrival at IST, what are my chances of getting into a TK lounge? I believe my best shot is the ‘Business’ lounge - but do any more experienced travellers know the success rate of ‘cash paid upgraders’ being given access to the J lounges?

Also, to clarify, my inbound boarding pass shows ‘business class’ but my onward boarding pass to RUH is in economy.

Travelling today. Thanks again for any pointers/advice!