You're quite right that the UK has never made foreign birth or family membership a disqualification for succession to the Crown, however, unless they have converted, the Greek ex-royal family are disqualified for not belonging to the Protestant faith. Under
the Act of Settlement (1700), succession to the British Crown is reserved for those heirs of the body of Princess Sophia, Electress and Dowager Duchess of Hanover, who are Protestants.
the Crown and Regall Government of the said Kingdoms of England France and Ireland and of the Dominions thereunto belonging with the Royall State and Dignity of the said Realms and all Honours Stiles Titles Regalities Prerogatives Powers Jurisdictions and Authorities to the same belonging and appertaining shall be remain and continue to the said most Excellent Princess Sophia and the Heirs of Her Body being Protestants