Paper bags are not covered by the new law. I have been offered paper carrier bags at several outlets that used to use plastic bags.
I think the British (sorry English) law is cock-eyed. The French do it much better. When they banned one-use bags, the ban was total, no one-use bags at all. If you arrived at the supermarket without a bag and needed one, you had to buy a long-life re-usable bag costing at least 1 euro. After a few weeks, all but a very few people turned up with their own bags. The exception is clothes. Whenever I buy clothes at a French supermarket they are rolled up and placed in one of those small thin polybags like those used for loose vegetables. It keeps them clean and protects them from the damage that being put in a re-usable bag might cause.