British Passport - in the UK, British passports are issued by the HM passport office. In conjunction with the post office, it is possible to submit most applications at a number of branches. Since all passports are issued in the name of the queen, the queen herself IS exempt from having passport to travel.
To apply for indefinite leave to Remain or #BritishCitizenship with entitlement to a British passport, you will need to demonstrate the knowledge of Life in the UK. From 26th October 2013: the rules have changed again. You will have to pass LIUK and prove the English efficiency at or above Entry 3 i.e. B1.