Killing Nandina Domestica
To get rid of Nandinas, The Maryland Invasive Species Council recommends to: "Spray with glyphosate or triclopyr during the winter, when other plants are dormant. Care should be taken when using glyphosate, because it is a broad-spectrum herbicide, and could harm non-target plants." If I do this, when would I be able to plant another shrub in that same spot?
As soon as the shrub dies, you can replace it. Glyphosate breaks down quickly, but you aren't spraying it on the soil or other plants anyway. If concerned about glyphosate, you could also just dig up the plant.