Elizabeth did not have a middle name. Her full name was Elizabeth Tudor and she was known as Princess Elizabeth until she was made illegitimate in 1536 when her mother, Anne Boleyn, was executed and the marriage between Anne and Henry VIII annulled. She then became known as The Lady Elizabeth.
Kings and queens of England go by their first names. Elizabeth, of course, was Tudor. When she became queen, she was Elizabeth R, meaning Rex, or queen. The current successor to the throne is Charles, whose last name is Windsor. His son, William, however, is known as William Wales, because his father is the Prince of Wales. When he succeeds to the throne, he will be William Windsor. The English are very fluid in their use of surnames.
Rex means king, for a woman, it’s Regina – so Elizabeth R = Elizabeth Regina.