When the Duchess of Sussex gives birth to her and the Duke of Sussex’s first child in the spring of 2019, the child will be born with United States citizenship through Meghan, as she is expected to still hold her American citizenship at the time.
Even though the child will be born on British soil, according to United States law and the State Department, he or she will be an American from birth, “A child born abroad to one U.S. citizen parent and one alien parent acquires U.S. citizenship at birth under Section 301(g) of the INA provided the U.S. citizen parent was physically present in the United States or one of its outlying possessions for the time period required by the law applicable at the time of the child’s birth.
“For birth on or after November 14, 1986, a period of five years physical presence, two after the age of fourteen, is required. For birth between December 24, 1952, and November 13, 1986, a period of ten years, five after the age of fourteen, is required for physical presence in the United States or one of its outlying possessions to transmit U.S. citizenship to the child. The U.S. citizen parent must be the genetic or the gestational parent and the legal parent of the child under local law at the time and place of the child’s birth to transmit U.S. citizenship.”
However, he or she will not be the first royal born with American citizenship. When American Princess Grace (born Grace Kelly) gave birth to her and Prince Rainier’s three children (Princess Caroline, Prince Albert and Princess Stephanie) in Monaco, they were each born with US citizenship through Grace. However, Albert gave up his American citizenship at the age of 21.
More recently, Princess Leonore of Sweden was born in New York City in 2014, and as a result, she has held American citizenship from birth. Leonore is a dual US/Swedish citizen. She is the child of Princess Madeleine of Sweden and her American-British husband, Chris O’Neill.
Dual citizenship is not unchartered waters for the British Royal Family. Princess Anne’s son, Peter Phillips is married to Canadian Autumn Phillips. Together, they have two daughters, Savannah and Isla Phillips, both of whom are dual Canadian/UK citizens. Savannah is 15th in line to the throne, while her little sister, Isla is 16th in line. As such, they are the first and closest Canadians in line to the British throne.
In the case of the child of the Duke and Duchess of Sussex, he or she will be a dual US/UK citizen, due to his or her birth on British soil and an American mother. This means that the new royal baby will be the closest (although not the first as Maud Windsor was born in California and currently ranks 50th in the line of succession as the granddaughter of Prince Michael of Kent) American in line to the British throne, as the child will rank seventh behind his or her grandfather, uncle, three cousins, and father.
You can read a comprehensive list of royals who were born with US citizenship here.