Incorrect actually.
The Act of Union did not address the nationality issue but the only distinction at that time was the common law distinction between aliens and subjects. Irishmen, like Scots were subjects of a single political entity, i.e. British Subjects.
The first codification of British Nationality law was the British Nationality and Status of Aliens Act 1914 which held that all the people of the empire whether Indians, English, Malays, Scottish, Welsh or Kenyans etc held equal status as British subjects.
As the empire broke up the British Nationality Act 1948 changed the name to Citizen of the United Kingdom and Colonies (CUKC)
Under the British Nationality Act of 1981, all the people of the United Kingdom including Northern Ireland are officially and correctly known as British Citizens.