Remember that you have a duty to get rid to an authorised scrap place. For a car, that means an ATF (Authorised Treatment Facility). There should be a list on the Environment Agency website somewhere. A DVLA logo on their adverts doesn't necessarily mean they're registered with them, and neither does "all paperwork taken care of" on a Gumtree listing.
Be sure to get a Certificate of Destruction from whoever picks it up - there and then, or within 24 hours usually. An ATF has access to the DVLA computer system, so they should be able to issue a CoD as soon as they've logged your car on as dead.
Otherwise the car remains your responsibility. e.g. if it gets dumped in a quarry or something, you'll have to go and get it and take it to a proper scrappy, plus you'll probably get nicked.