This is a pretty simple solution. NW had the bigger footprint in Asia. Delta will tie into NW's partnerships, which means you'll have a choice of JCB, Citi, or Dinners, all of them you can currently sign up with today. Just look at the NWA japan web site. You talked to the Singapore call center, which is actually NW's Asia call center. I'm not surprised they didn't have the info from Delta. This is probably the smaller issue they've created for themselves.
So what does this mean to you? If all you want is a credit card for Japan sign up for which ever card fits your needs best from the NWA japan site. All of them are SkyMiles cards.
On the other hand, if you want miles out of it then wait. Delta will sell your name to one of the card companies and you'll likely get some bonus miles for enrolling.
Rest assured, Delta will not be sending you a new credit card. They are not a bank. They have no authority to just send you a new card.
As far as DL messing up in Japan. I'm not surprised. There have been a number of mis-steps in the market. This is certainly not the last.