When your car is written off, we (insurers) do not register an interest in the vehicle at all - we notify the DVLA that the car has been passed to a motor trader (ie a salvage yard usually).
If acar is categorised as an A or B write-off, we usually just dispose of the vehicles as they should (and I use the word should here) not appear on the public road as they should be broken from scrap - VOSA are idiots in this respect, but that's another story.
This is probably the reason rugeleyboy that the DVLA think you no longer own the van, but don't quote me on that.
Equally, we do not ''own'' the vehicle until settlement has been agreed with the policyholder, and so legally, we cannot/should not dispose of the vehicle until this is agreed. As noted though, if the car is a burn out, or cut to pieces by the fire brigade, it's generally accepted that the car gets disposed of asap as to avoid any storage charges.
You're best not informing the DVLA, because we normally inform them anyway.