We have had these bay windows in our house just taunting me. My wife is very pleased with the outcome. She made the cushion and pillows.
I used Kreg screws for the 2×4 frame, and wrapped it with 1/2" birch plywood. Made the moldings from pine doing some bad things on a table saw for the molding pieces between the angles. The doors are rimmed with pine window casings and backed with some pine shoe moldings so the doors are flush.
With the mix of wood and knowing both don't stain well, I used pre-conditioner and then followed with a gel stain. For the pine pieces I had to use spray urethane otherwise the stain just came off. For the birch ply, I used Arm-R-Seal. I am pleased with the outcome.
For me it was a short project, it only took 4 months