Frankau's first television appearance was in an episode of the BBC's Sunday Night Theatre (1954); she acted in Emergency-Ward 10 and The New Adventures of Charlie Chan, both in 1957. Other early appearances include roles in The Man Who Finally Died (1959), No Hiding Place (1962), Six Shades of Black (1965), You Can't Win (1966), Intrigue (1966), Callan (1967), The Dustbinmen (1970), General Hospital (1973–75), Within These Walls (1975), Robin's Nest (1977), The Duchess of Duke Street (1977), Yes Minister (1981), I Remember Nelson (1982), Nobody's Hero (1982), The Cleopatras (1983), Mitch (1984), No Place Like Home (1984–86), Terry and June (1979-1987), Bergerac (1987), Boon (1990), 2point4 Children (1993), and Big Women (1998).
A former member of the Royal Shakespeare Company, Frankau played Hippolyta in A Midsummer Night's Dream in 1962 at the Royal Shakespeare Theatre. The play was directed by Peter Hall and starred Judi Dench and Ian Richardson.
Frankau's grandfather was Arthur Frankau, son of Joseph Frankau, a German Jew who came to London from Frankfurt in the late 1830s and started a cigar trading business. Arthur Frankau married Julia Davis, who was a celebrated novelist of satirical books. Some of her titles are Pigs in Clover and An Incomplete Etonian.
Her father was the actor Ronald Frankau, and her mother the actress Renee Roberts who played Miss Gatsby in Fawlty Towers. Her uncle Gilbert went into the family trading business until the war (living and working for a while in Germany). He was also a war poet and subsequently a novelist. Her half-brother is the TV producer John Frankau, father of the actor Nicholas Frankau. She also has a sister, Roberta.
Frankau married firstly Edward J. Fosh in 1953. They had no children and the marriage was later dissolved. In 1968 she married Bill Bain (died in 1982), an Australian TV director. They had two sons, Matthew (born 1969) and Sam Bain, a comedy writer who co-created Peep Show. She married Albert D. Bird in 1997 and the couple live in Kent.