Hi, it would be obviously helpful to buy the death printout for William, who, as he predeceased his wife, hopefully had in her an informant that knew e.g when and where he was born, and who his parents were (including his mother's maiden name), and, cross your fingers, all this will be recorded on the death printout. In 1919 in Invercargill I have Isobel with two of the children you mentioned at the same address, Robert Alexander, a returned soldier, and Elizabeth Margaret, spinster.
Craig.