Hi Sarah,
I have two death certs from ww2, one was the army and the other the airforce. You will find them in the section Deaths overseas 1939-1945 on 1837 online. The army one I sent for says Return of the warrant officers. Non commissioned officers and Men of the Royal Artillery, Killed in Action or who have died whilst serving aboard between 3rd sept 1939 and 30th June 1945.
Also it will tell you his Rank, Number (which helps in any other research) what he died of, weather he was killed in action, in the case of my Raf chap his cause of death is listed as Air Operations.
Also it will tell you where he died.
So I would think that the army registers the death and not the families.
I don't remember seeing an original death certificate when I borrowed my uncle's stuff from another relative, but there was the telgram informing the family of his death and a letter from the King. wouldn't be sure weather the letter from the king came after the war or not. so with all that I don't suppose family's reg deaths themselves, not sure but hope this helps. give 1837 online a try worth a fiver if you find him
best wishes
San
