When my old Wii U was replaced after being stolen and Nintendo did an Nintendo Network ID transfer, I found I couldn't access the shop or download patches, but the web browser and system updates still worked fine. I had to call them to get it fixed, since apparently the eshop account is handled separately behind the scenes to the NNID.
Thanks for the info.
Damn, that sure is complicated. I created a User on the Wii U and verified the email and everything. Linked my account online with the PC and all but I still can't access the eshop. Funny thing is that pretty much everything else works just fine. I even collected some coins by completing various tasks such as login into the Miiverse thing.
I also tried redeeming my code here: https://eshop.nintendo.net/redeem#/ It gives an error and says that my account isn't linked. On the settings it shows that I have linked it. Weird. Maybe the eshop is going through some problems.