The first permanent English settlement in North America was Jamestown, Virginia, founded May 14, 1607 by the Virginia Company of London. "British" is slightly anachronistic—the 1707 Act of Union created Great Britain; before that, it was English (and Scottish) colonization.
English/British colonies spread continuously along the Atlantic: New Hampshire, New York (taken from Dutch 1664), New Jersey, Pennsylvania, Delaware, North/South Carolina, Georgia (1733). By 1775, 2.5M people in 13 colonies.
Norse (Vinland, ~1000), Spanish (St. Augustine 1565, Santa Fe 1610), French (Quebec 1608), Dutch (New Amsterdam 1625) all preceded or paralleled English efforts. But Jamestown began the continuous Anglo-American line.