The Portuguese and the Spanish made the first official claims to lands in the New World through the Treaty of Tordesillas in 1494. The treaty was ratified by the Pope, and Portugal and Spain split the continents of North and South America among themselves. Primarily through the claims of Vasco de Balboa, Spain took the lands bordering the Pacific Ocean; Portugal concentrated on Eastern South America (Brazil).
The British, naturally, disputed these claims and founded the settlement of Jamestown in 1607. The British had previously laid claim to North America following John Cabot's discovery of Newfoundland in 1497, believed to be the second exploration of the New World after Christopher Columbus' voyages.
The French first claimed lands in North America following the voyage of Giovanni de Verrazzano in 1524-1525. Verrazzano explored the area between New York and North Carolina. Jacques Cartier claimed land for France during his voyage to the St. Lawrence River in 1535. The French solidified their claims with the founding of the colony of Acadia (Nova Scotia) in 1605.
I do not think your answer is going to be as cut and dry as you expect because they arrived at different places at different times and controlled those areas for different amounts of time.
The French were in Canada around 1534 but didn't colonize until about 1605. From there they began losing parts of Canada to the English (1713ish)
And they were also in the Gulf Coast in 1683 initially but didn't colonize until 1698.
The Spanish have Columbus in 1492 but didnt actually make their first land expedition until 1513. This is when they also came to Florida and controlled that until 1763 when the English took over for 20 years and then the Spanish took over again in 1783 until 1821.
The English attempted colonies in the Caribbean 1604 but failed. Jamestown was 1607. Obviously they lost control in 1776 with the Revolutionary War. Unless you actually consider the end of the war as loss of control then its 1783.