The problem has sort of solved it self, so it's no longer an issue. 🙂
I think that Apple has "done something" in the Appstore, but i noticed the following prior to the problem being solved, which might be an inspiration to others:
A similar problem existed in the Podcast application as items i subscribed to, wouldn't download when i pressed the "Down" button, that usually starts a download. Shutting down Podcast and restarting it, usually solved the problem after 2-4 tries.
Within Appstore in the "Purchased" section where you can see all apps downloaded, the problem with update is the same as in the "Update" section, described earlier.
I then tried to "disconnect" my Apple ID, and within Appstore start downloading some random free app, at which ID and password is requested, and after entering that, the download proceeded without problems.
While still in Appstore (and the entered ID + password still "active"), i then in the "Purchased" section, i could suddently download the updates waiting, one at a time, but with the "twist" that i in most cases had to try 2-5 times before the download started.
After that updates started working again, but still i had to try several times, and then after a week or so, it was back to normal where updates starts downloading immediately - problem solved.