The 'Premium' page offers 'accelerators' for building construction. I tried buying 1, but I can't tell if the accelerator was applied. Is it applied to all planets I own? Is there a status that reports if an accelerator is in effect?

When you buy a accelerator you have "it in your inventory".
See here: http://prntscr.com/kuss6y
 
After you bought the acccelerator you can go to any planet where you build any building. You then apply the accelerators by pressing the green or red buttons.
See here: http://prntscr.com/kussuz
 
When you do so the building time of this specific building is reduced.
They are not automatically applied when you purchase them.

Roger that, thanks for the clarification, Tii.