In 1953, he participated in the coronation of Queen Elizabeth II as a Page of Honour for his grandfather and carried his grandfather's coronet during the ceremony.

In 1965, Clyde appeared in Passion Flower Hotel, a stage musical written by John Barry and Trevor Peacock, at The Prince of Wales theatre, London. It also featured Jane Birkin, Francesca Annis, Pauline Collins, Nicky Henson and Bill Kenwright.

His film career has included roles in The Great St Trinian's Train Robbery (1966), Silver Bears (1977), North Sea Hijack (1980) starring Roger Moore, Invitation to the Wedding (1983), Wilt (1990), Splitting Heirs (1993), The Musketeer (2001) and The Iron Lady (2011).

In 1969, he appeared in Conduct Unbecoming as part of the original cast, which included Paul Jones. He also travelled to the US as part of the original Broadway cast.

Clyde is divorced from Vanessa Field, whom he married in 1970. They have two children.

Clyde once guest-starred in an episode of the American sitcom My Three Sons, when Chip Douglas is excited that someone from Liverpool was coming to visit and expected him to be a talented musician, implying the success of The Beatles. (The episode aired during the height of Beatlemania.) He appeared in the BBC TV adaptation of Moll Flanders in 1975, and in 1979 he played Godfried Schalcken in the BBC's television horror story Schalcken the Painter. He is perhaps best known for his portrayal of villainous Austrian Imperial Governor Hermann Gessler in the 1980s action series Crossbow, which incorporated Clyde's ability to convey evil in a distinctly aristocratic way. His other notable acting role was as Dick Spackman in the ITV sitcom Is it Legal?. Clyde also portrayed King Charles I in the BBC series By the Sword Divided (1983–85), which focused on the English Civil War (the beheading of the king is featured in the second episode of Season 2). Clyde also starred as Algernon Moncrieff in 1985 in the Great Performances production of Oscar Wilde's The Importance of Being Earnest opposite Gary Bond as Jack Worthing and Dame Wendy Hiller as Lady Bracknell.

In 2002, Clyde appeared in The Falklands Play (a BBC dramatisation of the Falklands War) as Sir Nicholas Henderson, the British Ambassador to the United States at the time. In 2004, he appeared in the BBC drama series The Alan Clark Diaries as British Conservative Politician Jonathan Aitken and also appeared in the BBC drama series Ashes To Ashes as the Superintendent which was aired in 2008.

In 2017 he played Dennis in The Girls at the Phoenix Theatre in the West End.