[DanaC]

Nope. Legally we are citizens. There's a very small number of people who are legally 'subjects'.

On 1 January 1983, upon the coming into force of the British Nationality Act 1981, every citizen of the United Kingdom and colonies became either a British citizen, British Dependent Territories citizen or British Overseas citizen.

Use of the term British subject was discontinued for all persons who fell into these categories, or who had a national citizenship of any other Commonwealth country. The category of British subjects now includes only those people formerly known as British subjects without citizenship and people born in Ireland before 1949. In statutes passed before 1 January 1983, however, references to British subjects are interpreted as if they referred to Commonwealth citizens.

British citizens are not British subjects under the 1981 Act. The only circumstance where a person may be both a British subject and British citizen simultaneously is a case where a British subject connected with Ireland (s. 31 of the 1981 Act) acquires British citizenship by naturalisation or registration. In this case only, British subject status is not lost upon acquiring British citizenship. The status of British subject cannot now be transmitted by descent, and will become extinct with the passing of all existing British subjects.

British subjects, other than by those who obtained their status by virtue of a connection to Ireland prior to 1949, automatically lose their British subject status on acquiring any other nationality, including British citizenship, under section 35 of the British Nationality Act 1981.

In 2010, around 3,500 British Subject passports were issued each year, with the number steadily declining over time.[10]

Though the term British subject now has a very restrictive statutory definition in the United Kingdom—and it would be incorrect to describe a British citizen as a British subject—the concept of a subject remains in the law, and the terms the Queen's subjects, Her Majesty's subjects, etc., remain in use in British legal discourse.[11]

Incidentally, here's John Oliver's take on the matter: