I obviously hadn't tried everything I could think of...

I got it to work now. 

I entered the bluetooth-settings on the phone when the headphones was not connected. Then I removed/disconnected the connection between the phone and the headphones. So the next time I connected the headphones to the phone, it was as if I connected the phone to the headphones "for the first time". I then got a notification about the google assistant. I turned on notifications/permissions for google, and then it worked. (This was after I changed the NC-button" in the headphones app.