When my dad passed away a few years back, I phoned the council and was put through to the 'bereavement department'. I had expected to have to notify a number of different departments with which dad had dealings, but this department said that, to make things easier for us, this one phone call would suffice and they would deal with all the other various departments as necessary. They requested a copy of the death certificate but after that everything from blue badge to council tax, attendance allowance etc were dealt with without any more input from us.
Some seem to be able to get it right.