Jennings was born in Essex, the son of Peggy Patricia (née Mahoney) and Michael Thomas Jennings. He attended Abbs Cross Technical High School in Hornchurch and then studied English and Theatre studies at the University of Warwick, graduating in 1978. He said he saw his first theatre when he was in high school and went to the Old Vic Theatre, which led him to be inspired to be an actor.

Jennings began his career in regional repertory theatre, in 1985 playing a range of roles including Maximilien Robespierre in The Scarlet Pimpernel. Jennings met Director Nicholas Hytner during this production and has worked with him many times since.

For his performance as Gloumov in Too Clever by Half at the Old Vic, he won the Olivier Award for Best Comedy Performance in 1988. He was nominated in the same category the following year for portraying Dorante in The Liar.

His Royal Shakespeare Company roles include the title role in Peer Gynt (for which he won an Olivier Award 1995-06 for Best Actor), the title role in Richard II (opposite Anton Lesser as Henry Bolingbroke), Theseus/Oberon in A Midsummer Night's Dream (UK, American tour and Broadway), Angelo in Measure for Measure, and the title role in Hamlet. I

In 2002, he appeared in the Cameron Mackintosh/Trevor Nunn revival of My Fair Lady at the Theatre Royal, Drury Lane and won an Olivier Award as Best Actor in a Musical. He was an Associate Artist at the Royal Shakespeare Company.

In 2007, he portrayed the Rev Hutton in the BBC series Cranford. He also played John Le Mesurier in the one-off BBC drama Hancock and Joan.

He has recorded the audio versions of the books The Horse and His Boy, Out of the Silent Planet, and Perelandra by C.S. Lewis, 20,000 Leagues Under the Sea by Jules Verne, and Attention All Shipping by Charlie Connelly, which was selected in June 2008 as one of the top 40 audiobooks of all time. In 2006, he recorded an abridgement of A Spot of Bother by Mark Haddon. He is also a regular narrator on BBC Radio 4's Book at Bedtime.

In 2009, he appeared in The Habit of Art as Benjamin Britten.

In 2010, he played Captain Shipshape in the CBeebies second series of Grandpa In My Pocket and starred in the film Belle. After that, he played Henry Tizard in Castles in the Sky.

From 2011 to 2014, Jennings played Alan Cowdrey QC in the BBC One legal drama Silk. In 2016, he appeared in the Netflix series The Crown as Prince Edward, Duke of Windsor (former Edward VIII), granduncle to Prince Charles whom Jennings played in The Queen), as well as Leopold I of Belgium in the ITV series Victoria.

In 2014, he played the role of Willy Wonka in Charlie and the Chocolate Factory the Musical, which was directed by Sam Mendes and was performed on London's West End theatre district. He took over the role from Douglas Hodge in 2014.

Jennings portrays Playwright Alan Bennett in the 2015 film The Lady in the Van opposite Academy Award winner Maggie Smith in the title role. The film is directed by Nicolas Hytner, who is a long-time collaborator from the theatre.

In 2016, he reprised his role as Professor Henry Higgins in the Australian 60th Anniversary production of My Fair Lady, directed by Julie Andrews.