I've decided to post my progress on Mollie's "custom stroller", and also instructions in case any one wants to try it.
The first step is to remove the stroller seat and storage compartment from the stroller's frame. Mine were attached using D-rings and snaps, so it was easy to take apart. If yours pieces are sewn on though, you can just cut them off!
Next, you need to use a seam ripper to rip out any seams or darts in the fabric, so that the piece lies flat. It's a good idea to take pictures as you do this, and mark any dart lines on your piece so you know where to put the darts in your new seat.