In 1973 she made her professional stage debut, in Bristol, England, while appearing in a regular role in the British television series The Onedin Line. In 1974 she was invited to London to play the part of Jenny in David Hare's play Knuckle at the Comedy Theatre, followed by a season with the National Theatre Company playing Ellie in Heartbreak House. 1975 saw her appear opposite Anthony Hopkins in the televised play The Arcata Promise followed by the televised theatrical version of The Count of Monte Cristo that featured an all-star cast of British and American actors. That same year her first feature-length film The Romantic Englishwoman was released.

In 1977, again with the National Theatre, she gave a "stunning" performance as Marianne, opposite Stephen Rea, in Tales from the Vienna Woods directed by Maximilian Schell. Also in 1977, she played the part of Rosalind in As You Like It, directed by Terry Hands, opposite Peter McEnery in Stratford-upon-Avon and the following year in London. This she followed with Plenty, another play from David Hare, at the National Theatre, for which she received a 1978 Olivier Award nomination for Best Actress in a New Play. The winner was Joan Plowright in Filumena. She was cast in a similar role, playing opposite Bill Paterson, in Hare's BAFTA-award-winning companion play Licking Hitler, for BBC television.

Again on screen, in 1978 she played the part of Isabella in the BBC Television Shakespeare production of Measure for Measure, a performance that led the New York Times to describe her as providing "the image of idealized faultlessness". In 1979 she was the female lead alongside Frank Langella and Laurence Olivier in Dracula. In 1981 she starred opposite fellow Canadian Donald Sutherland in Eye of the Needle, a wartime espionage thriller based on the Ken Follett best-selling novel. Two years later Nelligan moved to New York City, where she earned nominations for the Tony Award as Best Actress in a Play three times and one Featured Actress in a Play between 1983 and 1989 for the five Broadway plays in which she appeared.

In 1991 she won a BAFTA for "Best Actress in a Supporting Role" for her performance in Frankie and Johnny. For her performance in the 1991 film The Prince of Tides she was nominated for the Academy Award for Best Supporting Actress. Between 1987 and 2004 she was nominated for five Gemini Awards for her performances in Canadian television mini-series and films and won twice, both in 1993. In 1996 she played the wife of James Mink in the CBS television film Captive Heart: The James Mink Story.