Cannot add any card

I cannot add any card to my wallet. It always ends with: Try again later or contact your card issuer for more information.

I was able to click on '+' or 'add card' button, scan the card or simply put there the card information manually, then a form from correct card issuer was present to me. But after agreeing on card issuer terms I always get the before mentioned: try again later or contact card issuer.

I already tried reset of the phone, reconnecting to Apple ID etc. Nothing helped.

The problem is not in the card themselves either as it was possible to add the same cards on my wife's iPhone.