He also announced Rachel Noble,who is currently responsible for the federal government's cyber security strategies,will lead the Australian Signals Directorate from February next year.
"Her appointment to this leadership role is a significant step forward for women in the national security sector and we congratulate her,"Mr Morrison said.
In announcing Dr Nelson's successor,the Prime Minister said Mr Anderson,having written three children's books about Australia's involvement in two world wars"is well placed to guide the Australian War Memorial through generational change to tell the stories of our more recent conflicts".
"He has demonstrated his passion for military history and ensuring future generations are educated about the service and sacrifice of our personnel."
Mr Anderson spent eight years as an Australian army officer after graduating from the Royal Military College at Duntroon.
Prior to his London posting,Mr Anderson served as ambassador to Afghanistan.