I had he same issue and the following fixed the button doing nothing for me:
Removed the iphone app
Reinstalled it
Logged in again
Enabled new download feature
Restarted app (just to in case)
Clicked download
This time when I clicked download I got a temporary notification at the bottom of the screen saying Download Scheduled.
Now I see in the main menu:
Downloads: Downloading 1 of 15 but its been sat there for minutes no progress. If I click into that I see the first track listed as Downloading… and the summary at the top of 0.0Kb 1.
Seems the new download method isn’t ready for release.