Mark the rows or leave a bare path between them so that you don't tread on the tubers before they come through. Continue to top up the mulch as the shoots grow. Cover the whole area, including the paths. You can top this up as needed, provided you allow the first layer of mowings to dry off before applying a second. Slugs are no more of a problem with this method than any other.
Blackbirds can be a bit of a nuisance as they pull the mulch about.
Make sure that you replace the mulch carefully afterwards. Early potatoes can be ready in as little as eight weeks from planting. For maincrop varieties, wait until the foliage has died down before harvesting. If you want to harvest the whole crop it's easier to remove the mulch, collect the tubers and then replace it. Maincrop varieties should be picked on a sunny day, if possible, and allowed to dry out on the surface for a few hours before putting into bags.