Hi Andrew,
When doing an initial mulch with a thicker dressing of, say, four inches of compost and manure, I like to plant straightaway. Unless you have rampant bindweed or a Spanish equivalent, I would plant potatoes now, through small holes in cardboard and plastic. Check after 3-4 weeks that potato stems are managing to push up through the holes, and they may need helping to do so. Once plants are growing, their tubers will stay white as the mulch is acting to keep light off them, in the same way as soil in ridges.
You could try a second crop from planting in late summer, after harvesting by pulling back the mulches.