I forgot to mention that the washing machine is never allowed to agitate. All you use it for is to soak and to spin. All agitation is done by hand, pushing down on the quilt from above. (Obviously won't work in a front loader!)
If you don't want to tumble dry the quilt, it needs to be laid out flat on a clean sheet. Outside on the grass is fine *but* it must be in the shade (light, including sunlight, fades fabric) and it must be covered with another sheet (to prevent damage from bird droppings). You never want to hang a wet quilt over clotheslines; with the weight of the water in the quilt, this puts enormous stress on the seams and fabric.
For a nice antique quilt like this, I would dry only until lightly damp and then block on a flat surface.