But the book claims the Queen's husband, Prince Philip, was rather less enthusiastic about the situation.
It says Prince Philip thought Harry and Meghan were doing the wrong thing in giving up royal duties to move to the US.
"It's a big mistake to think about yourself ... If you think it's all about you, you'll never be happy," the book claims he said.
Earlier extracts from Brandreth's book have delved into the Queen's final months, claiming she found solace watching TV show Line of Duty following her husband's death, and that she reluctantly accepted she needed to cut down her schedule as she became more frail.