Some types of plastic films can be recycled at carrier bag collection points at the larger stores of most major supermarkets - see list at “Carrier Bags” (Recyclenow.com, 2019). Collection points for used carrier bags are normally found near the main entrance. For online shopping deliveries, many supermarkets also allow you to hand your unwanted bags back to the driver for recycling.
The plastics submitted at a supermarket don't have to be that supermarket's packaging. As a general rule, if you can stretch the film then it can be recycled. This includes films like bubblewrap, but not compostable and biodegradable bags which can potentially cause quality issues in the recycled material. For a specific list of plastics that can or can not be recycled at supermarkets, see “Plastic Film & Carrier Bags” (Recyclenow.com, 2019).
M&S accepts a larger variety of plastic packaging, including black ready meal trays, crisp packets, sauce sachets and certain cosmetics containers - “M&S TO TURN PLASTIC INTO PLAYGROUND EQUIPMENT WITH NEW PLASTIC TAKE-BACK SCHEME” (M&S, 2019).
Tesco is running trials in and around Swindon and Bristol to recycle all soft plastic packaging, including pet food pouches & crisp packets. List of stores is available at “Tesco Trials Technology That Makes All Plastic Packaging Recyclable” (Tesco PLC, 2019).
Many supermarkets are also running plastic bottle recycling trial schemes.