The last Royal to hold any form of political authority in England was George III, and he used that authority to pass the Royal Marriages Act in 1772. Since then, as part and parcel of the Family being above politics, no Royal may express interest in, stand for or hold any public political office.
The Royal Family stand for neutrality in the United Kingdom and are a banner to unite behind when the country may otherwise be divided by politics. It would therefore be inappropriate for one of its members to serve at the behest of Government by either party. There is some rumor that it was this restriction, rather than just his marriage to Wallis Simpson, that persuaded Edward VIII to abdicate the throne; he felt unable to hold his tongue or refrain from giving his opinion to the Prime Minister of the time.