I don't know how easy these would be to upholster, I think all the spindles are connected to the seat right? I would buy or make tie-on chair pads.
That's what I'd do also though it would be possible to cover the seats, you'd have to cut slits in the fabric to go around the spindles in the back. Very time consuming.
If you are an excellent pattern maker and seamstress, this can be made and then stapled onto the chair underneath. For the average person, a beautiful chair pad is the answer. You can either make one or take it in to have a custom pad made according to the thickness you want, the style, the fabric, etc. If you make a cover, you can put a lot of batting at the back, have a couple of sewn in finished off strips to go through every spindle and then staple it underneath. Then finish underneath by hot gluing on a cover over staples.
Another option would be leather, set with padding and upholstery tacks. We have an old antique corner chair that was done that way.
I picked up some tacks on ebay for the padded areas of my lap desks.