They will keep all your details on file for a certain number of years.
Just tell that that you want no marketing literature sending, I'm sure there is something they can put against your account to stop it, NatWest certainly can!
Why don't you 'go green' instead of closing your account? I have at HSBC so I don't receive any paperwork any more. I don't know if your bank does this but they should. As for what information they keep if you do close the account I'm not sure.