Her Majesty is quite certainly of British heritage. She’s not descended from all of the kings and queens that preceded her on the throne, and holds the throne by virtue of an Act of Parliament from three centuries ago, but she definitely is of British heritage.
Elizabeth II claims the throne by being the leading descendant of the Electress Sophia, whom Parliament chose as the heir to succeed Queen Anne in the Act of Settlement of 1701. Sophia in turn was the grand-daughter of King James I of England. Through James I, Her Majesty is directly descended from William the Conqueror and most of the English kings that held the throne between 1066 and 1625, the year James I died. (The prime exceptions from the English monarchs prior to James I are that she’s not descended from the Lancastrian kings, nor from any of the Tudors other than Henry VII.)
In fact, Her Majesty’s royal ancestry pre-dates William the Conqueror. One of her ancestors, King Henry I, married a princess of the pre-Norman, English royal house, which means that she can trace her ancestry back into the various Anglo-Saxon kings, including Alfred the Great.
Since James I was also King James VI of Scotland, Her Majesty is also directly descended from all of the Scottish monarchs in the Stewart line, with links to the pre-Stewart Scottish monarachs as well.
She is also directly descended from all but three of the kings and queen that held the combined throne of the United Kingdom from 1714 onwards, when the first Hanoverian, George I, her many-great grandfather, became king. (The three exceptions are George IV and William IV, her many-great-uncles, and Edward VIII, her uncle.)
She is not descended from the Stuart monarchs other than James I/VI. That would be Charles I, Charles II, James II, William III, Mary II, and Anne, who reigned between 1625 and 1714.
Considering that her direct ancestors have been on the throne for approaching 300 years, and that she traces her lineage back to most of the English and Scottish monarchs prior to 1625, it seems a bit academic to say that the Windsors are not really British.
(And as for Elizabeth II herself, considering that her mother was Scottish, daughter of the 14th Earl of Strathmore and Kinghorne , her British ancestry is pretty well established.)
Interestingly, the marriage between the Prince of Wales and Princess Di has added a bit more Stuart blood to the mix. I recall seeing at the time of the marriage that the Spensers are descended from one of Charles II’s “cough” natural sons, as well as from one of James II’s natural sons.