The fabric just pulls off and, if you're lucky, leaves the sponge backing behind.
Use the original fabric as a template and cut your material *exactly* to it.
Use thin, slow-drying glue for most of the material to allow repositioning. Not PVA as it needs porous material to air dry. Evostick or similar should be OK.
Use fast drying glue (superglue) for the edges. Have some acetone based nail-polish remover and a non-linty cloth to clean up mistakes (quickly, before it really sets, and carefully as acetone can melt plastics. If you have to use it use it quickly on as small an area as possible, then wipe it over as big an area as possible to spread it out and blast it with a hairdryer to evaporate it before it can do more damage. Spreading it out also means it blends any damage over a larger area and makes it less noticable)
Keep a rag with some nail polish remover on it to hand and wipe your fingers whenever you use the superglue as it's easy not to notice you have some on your fingers and it dries white and makes a right mess.
Work the edges in a bit at time working from top (exposed) to bottom (more hidden by handles etc so creases/gaps are less noticeable).
Use something flexible and non-porous to push the edges down into the outer gully. I used a 1x10" strip of thin corrugated cardboard which worked well but flecks of it got stuck to the glue. A cheap, bendy, soft plastic 12" ruler would be ideal.
Take your time and be careful, one drip of glue in the wrong place can ruin everything.
Took me about and hour-and-a-half for both front door but I was rushing and not being very careful.