British Nationality could be obtained through your birth, or through the birth of any one of your parents or grandparents, in Israel. This arises because of Britain's relationship with the Middle East and Britain's Colonial history.
To determine what form of British Nationality you might have, an analysis of your family's history must be done, and then applying it to the Nationality laws governing Israel that have been passed over the years.
STATUS OF ISRAEL
Up until 28.09.1923, the Crown exercised extra-territorial jurisdiction over the Palestine (which included what is now Israel).
On the 29.09.1923, Palestine became a British Mandated Territory.
Both the mandate and the jurisdiction ended when the State of Israel was established on 15 May 1948. From that day and up to the present, Israel is considered as a foreign country.
For more information on how British Nationality could be gained through your birth (or a parent's birth) in a British Mandated Territory coupled with a UK-born grandparent, watch this short video by Philip Gamble: