Born in Glasgow, Ure was the daughter of civil Engineer Colin McGregor Ure and Edith Swinburne. She went to the independent Mount School in York, where in 1951 she played the role of the Virgin Mary in the York Cycle of Mystery Plays, revived for the Festival of Britain. She trained for the stage at the Central School of Speech and Drama, then based at the Royal Albert Hall, London, where her classmates included the Actress Wendy Craig. In her final year, 1954, she won the Carlton Hobbs Bursary to join the Radio Drama Company, but declined it. Known for her beauty, Ure began performing on the London stage and quickly developed a reputation for her abilities as a dramatic Actress.

Ure was known principally as a stage Actress. She made her London debut as Amanda in "Time Remembered" (1954). She played a leading role as Alison Porter in John Osborne's new play Look Back in Anger (1956). In 1958, she was in the Broadway production of Look Back in Anger and earned a Tony Award nomination for Best Dramatic Actress. In this period, she also performed a season with the Royal Shakespeare Company at Stratford-upon-Avon and, while pregnant, performed in the 1960 London production of The Changeling at the Royal Court.

In 1956, Ure began an affair with married Playwright John Osborne while working on the initial production of his play Look Back in Anger. The couple married in 1957, had son Colin in 1961, and divorced in 1963. Osborne had continued having affairs during the marriage, and Ure started an affair with her co-star Robert Shaw in 1959, while the two were performing in the London stage production of The Changeling. It is believed that Shaw was Colin's natural father. Ure and Shaw married in 1963, with Shaw immediately adopting Colin. Ure and Shaw had three more children together, Elizabeth, actor Ian Shaw, and Hannah. Ure and Shaw were still married when she committed suicide in 1975.

Ure first appeared on screen when she transferred her fragile, captivating portrayal of "Alison Porter" from stage to screen in the 1959 film adaptation of Look Back in Anger. In 1960 she appeared in the film Sons and Lovers as Clara Dawes, earning nominations for both the Golden Globe Award and the Academy Award for Best Supporting Actress.

In 1963, after an absence of three years, she returned to film with a performance in the sci-fi drama The Mind Benders. She followed with roles in The Luck of Ginger Coffey (1964) and Custer of the West (1967), both with then-husband Robert Shaw. After 1968's Where Eagles Dare it would be five years before Ure's next and last film appearance, in 1973's A Reflection of Fear co-starring her husband.

Ure continued to perform on stage while performing in films over the next 13 years, but her growing alcoholism affected her stage career to the point that she was fired from the 1974 pre-Broadway production of Love for Love and was replaced by her understudy, Glenn Close.

Ure suffered from alcoholism coupled with a continued deterioration of her mental health through the early 1970s. On 2 April 1975 she appeared on the London stage with Honor Blackman and Brian Blessed in an adaptation of the teleplay The Exorcism, and after a disastrous opening night was found dead aged 42, from an overdose of alcohol and barbiturates. Her body was discovered by her husband Robert Shaw in their London home.