At the beginning of this book, the journalist father of Robert, Anthea, Cyril, and Jane has gone overseas to cover the war in Manchuria. Their mother has gone to Madeira to recuperate from an illness, taking with her their younger brother, the Lamb. The children are living with an old Nurse who has set up a boardinghouse in central London. Her only remaining boarder is a scholarly Egyptologist who has filled his bedsit with ancient artefacts. During the course of the book, the children get to know the "poor learned gentleman" and befriend him and call him Jimmy.
Cook's house is in Fitzrovia, the district of London near the British Museum, which Nesbit accurately conveys as having bookstalls and shops filled with unusual merchandise. In one of these shops the children find the Psammead. It had been captured by a trapper, who failed to recognise it as a magical being. The terrified creature cannot escape, for it can only grant wishes to others, not to itself. Using a ruse, the children persuade the shopkeeper to sell them the "mangy old monkey," and they free their old friend.
Guided by the Psammead the children purchase an ancient amulet in the shape of an Egyptian Tyet (a small amulet of very similar shape to the picture can be seen in the British Museum today), which should be able to grant them their hearts' desire: the safe return of their parents and baby brother. But this Amulet is only the surviving half of an original whole. By itself, it cannot grant their hearts' desire. Yet it can serve as a portal, enabling time travel to find the other half.
In the course of the novel the Amulet transports the children and the Psammead to times and places where the Amulet has previously existed, in the hope that â somewhen in time â the children can find the Amulet's missing half. Among the ancient realms they visit are Babylon, Egypt, the Phoenician city of Tyre, a ship to "the Tin Islands" (ancient Cornwall), and Atlantis just before the flood. In one chapter, they meet Julius Caesar on the shores of Gaul, just as he has decided that Britain is not worth invading. Jane's childish prattling about the glories of England persuades Caesar to invade after all.
In each of their time-jaunts, the children are magically able to speak and comprehend the contemporary language. Nesbit acknowledges this in her narration, without offering any explanation. The children eventually bring "Jimmy" (the "Learned Gentleman") along with them on some of their time trips. For some reason, Jimmy does not share the children's magical gift of fluency in the local language: he can only understand (for example) Latin based on his own studies.
In one chapter the children also come to the future, visiting a British utopia in which H.G. Wells is venerated as a prophet. Wells and E. Nesbit were both members of the Fabian political movement, as was George Bernard Shaw, and this chapter in The Story of the Amulet is essentially different from all the other trips in the narrative: whereas all the other adventures in this novel contain scrupulously detailed accounts of past civilisations, the children's trip into the future represents Nesbit's vision of utopia. This episode can be compared to many other visions of utopian socialist futures published in that era; Nesbit's is notable in that it concentrates on how the life of children at school would be radically different, with economic changes only appearing briefly in the background. (It seems somewhat akin to William Morris's News from Nowhere.) It also mentions a pressing danger of Edwardian England: the number of children wounded, burned, and killed each year. (This concern was addressed in the Children Act 1908, and later in the Children's Charter.)

Question: Where did the children stay while their parent's were gone?
Answer: They stayed at a boardinghouse in Central London run by an old nurse.
Alternative answer: A boardinghouse in Central London