Plastic bubblewrap does not biodegrade and is not recyclable. Reuse as much as possible.
Look for bubblewrap made from 100% recycled product, or eco-bubblewrap.
Reduce the need for bubblewrap by sourcing other packaging materials: shredded or scrunched paper, popcorn, fabric.
7 alternatives for bubblewrap are listed in this article by noissue.
Bubblewrap can be used to seal and insulate draughty windows in winter.
Look for craft ideas for bubblewrap on the internet.
Clean bubblewrap can be recycled with soft plastics in the receptacles at Countdown and The Warehouse.
Keep out of landfill wherever possible.