I am planning to recover this old chair with some new fabric so the first order of business was to remove the cushions.
I want to recover this chair.
The bottom cushion is simply held on with screws.
Two Phillips head screws on either side of the chair hold the seat on from underneath.
But the seat back is a little more mystifying. I guessed that the wood buttons on the side of the chair somehow hid the mechanism that held the top of the cushion.
These little wooden buttons are removable.
I popped off the buttons with the tip of a knife and sure enough there were screws that could be removed by an Allen wrench.
Loosen the screws with an Allen wrench (also called a hex key).
After removing the top screws the seat back was still not out of the frame. It swiveled but did not come out and I couldn’t see the any screws or buttons to hold it in.
After removing the top screws the chair back was still stuck in place.
I carefully spread the stiles and noticed the bottom of the seatback was held in by small dowels attached to the seat which were inserted into holes on each side.
A short dowel holds the bottom of the chair’s back pad in place.
I spread the stiles a little farther apart and one side popped out, then the other one.
Holes in each stile were revealed when the back was removed.
Mystery solved. (See the Before and After here.)
Are you better at taking things apart or putting them together?
I love your post and I did wonder myself how to dismantle a chair like this!
Look out for my chair redo post we can compare notes later. Can’t wait to see how you recover yours. Good luck with your re-do. 
Actually, it’s so funny that I stumbled upon your post today – as my project today is stripping back an old chair and recovering it!!
I’ll keep a lookout for your NEW chair. Mine will post in a few days. Jo