Thatcher was born in Bombay, British India, to English parents Torin James Blair Thatcher, a police officer, by his wife Edith Rachel, a voice and piano Teacher, younger daughter of the Hon. Justice Sir Herbert Batty, a puisne judge of the High Court of Bombay. He was educated in Britain at Bedford School and at the Royal Academy of Dramatic Art. He worked as a schoolmaster before first appearing on the London stage in 1927 and then entering British films in 1934. He appeared in the 1937 Old Vic stage production of Hamlet, in which Laurence Olivier made his first appearance in the title role, opposite Vivien Leigh as Ophelia. During the Second World War, he served with the Royal Artillery and was demobilised with the rank of lieutenant colonel.

Thatcher appeared in classic British films of the late 1930s and 1940s, including Major Barbara (1941) and Great Expectations (1946), in which he played Bentley Drummle. He moved to Hollywood in the 1950s. He was constantly in demand, invariably lending his looming figure and baleful countenance to sinister or stern roles in popular costume thrillers such as The Crimson Pirate (1952), Blackbeard the Pirate (1952), The Robe (1953) (as the disapproving father of Marcellus), The Black Shield of Falworth (1954), Helen of Troy (1956), Darby's Rangers (1958) and The 7th Voyage of Sinbad (1958). He was a formidable barrister in Witness for the Prosecution (1957) and also appeared in the Marlon Brando and Trevor Howard 1962 remake of Mutiny on the Bounty.

He returned to the stage quite frequently, notably on Broadway, in such esteemed productions as Edward, My Son (1948), That Lady (1949) and Billy Budd (1951). In 1959, he portrayed Captain Keller in the award-winning play The Miracle Worker with Anne Bancroft and Patty Duke. All of these plays were filmed, but Thatcher did not appear in the film versions.

Also a steady fixture on television, he appeared in such made-for-TV films as adaptations of A.J. Cronin's Beyond This Place (1957) and The Citadel (1960), Bonanza (1961), and Brenda Starr (1976). He also played the title role in a Philco Television Playhouse version of Othello and acted in a CBS production of Beyond This Place (1957). He appeared in programmes such as The Real McCoys, Thriller, Gunsmoke, Perry Mason, Voyage to the Bottom of the Sea, Daniel Boone and Mission: Impossible. He was cast as the sly space trader in an episode of Lost in Space. Thatcher, if not for his distinguished career, has become recognised by several generations of fans of Star Trek: The Original Series from his performance as Marplon, secretly a member of the resistance in the episode "The Return of the Archons".

Thatcher died of cancer on 4 March 1981, in Thousand Oaks, California, USA.