Henry's fears about heirs didn't come true just in his own lifetime. None of Henry's three heirs who ruled England in their turns -- Edward VI, Mary I, and Elizabeth I -- had children (nor did Lady Jane Grey, the "nine day queen"). So the crown passed after the death of the last Tudor monarch, Elizabeth I, to James VI of Scotland who became James I of England.
The Tudor roots of the first Stuart king, James VI of England, were through Henry VIII's sister, Margaret Tudor. James was descended from Margaret (and thus Henry VII) through his mother, Mary, Queen of Scots, who had been executed by her cousin, Queen Elizabeth, for Mary's alleged role in plots to take the throne.