I've just successfully repaired the fabric board part of the door cards on my Williams. I hear it's very common for the glue to dry up and the fabric comes away.

The boards are glued to the plastic door cards with some sort of yellow automotive glue. If you're lucky they will have dried out enough for the boards to be pulled free, starting from the front end, easily. Otherwise they will tend to pull on the board and they will slightly tear off the bottom later of the board they are glued to.

The fabric is glued to the board with a thin tacky sort of membrane layer. The side which is attached to the fabric seems to dry and disintegrate (from UV rays I guess) into a fine orange power, leaving behind a slightly spongy membrane on the board. If you turn the board upside-down you should see how the membrane neatly wraps around the whole board.

With some careful application of car-trim glue, it's possible to glue the fabric onto the remaining membrane and, any dirty age-related marks aside, make it look good as new from the top. You just need some double sided tape or something to glue it back onto the plastic door card; I've estimated that the pre-existing yellow glue (and the thickness of the wrapped-around membrane/fabric) is roughly just less than 1/8" thick, so I think some double-sided foam tape (the green stuff) cut into squares where the yellow glue was blobbed on will do the trick nicely.

I hope this helps anyone with the same issue. I'm lucky I work in upholstery so I have the tools at hand - If anyone wants there's done for them I'm happy to do it for them if you post them to me!