Copenhagen is an unusual birthplace for the granddaughter of a Scottish Earl and an English Duke. The reason was that her father, John Campbell, Lord Glenorchy, heir to the Earl of Breadalbane, was envoy extraordinary to the court of Denmark. Her mother was Amabel, eldest daughter of Henry Grey, Duke of Kent, of Wrest Park, and she was named Jemima after the Duchess. It was “a very easy and short labour” on 20 October 1722, wrote her father to the Duke. The Danes were considerate; their guards were ordered to go without drums or any noise when they marched to their parade ground nearby; and their commander put a chain across the road so that coaches must pass on the other side of the square. Yet it was not expected that Jemima would play other than a minor part in aristocratic life. She had an elder brother, Harry, born the previous year; and her uncle, Anthony, Earl of Harrold, was heir to the Dukedom of Kent.

The little household in Denmark kept by Lord Glenorchy and his young wife Amabel was a happy one. Their outward journey had had its mishaps in the saddest rainy weather that ever was seen; but this made the travellers as glad to get to a barn and a straw bed at night as if it had been a palace, and “we took your Grace's advice of making ourselves merry with all those little difficulties.” At first there were court functions to attend—drawing rooms every night. Amabel assured her father that she would study nothing so much as to please the people. (It was the Duke who carried on the correspondence; perhaps the Duchess was not much at home with the pen). When the court removed to Frederiksborg for the summer, about as far from Copenhagen as Windsor from London, there was some inconvenience in Glenorchy's having to go there twice a week, and so they took a country house. The gardens were well enough, and they had a great deal of wood about them, full of nightingales, where there was very good walking; but the house stood high and had constantly high winds.