Gillespie is the eldest child of Magdalena Katalin Singer, from Budapest, Hungary; and James william Gillespie, from Toronto, Canada. He was born in Lille, but the family left France in 1940 after Hitler's invasion of the country.

Gillespie was educated at Sale Grammar School, and trained as an actor at RADA between 1951 and 1953.

Gillespie began his acting career with two years with the Old Vic Company, beginning in autumn 1953 for Michael Benthall's Shakespeare seasons. In the company were Richard Burton, Clare Bloom, Fay Compton and Michael Hordern. The second year was headed by Paul Rogers, Ann Todd, Virginia McKenna and John Neville. Gillespie's first substantial part was Adam in As You Like It. The highlight of both years was Douglas Seale's production of Henry IV Parts 1 and 2. His first major TV role was as the disciple Matthew in Jesus of Nazareth, directed by Joy Harington.

Gillespie appeared in a string of popular British television series, mostly in the 1960s to 1980s. Credits include The Saint, The Avengers, Doomwatch, The Sweeney, The New Avengers, Survivors, Warship, The Professionals, Mary's Wife, I Woke Up One Morning, Return of the Saint, Bonjour La Classe and Secret Army. More recently, Robert has appeared in New Tricks, with Likely Lads' co-star James Bolam, as well as Jimmy McGovern's Broken, alongside Sean Bean.

By 1963 Gillespie was writing for Ned Sherrin's That Was The Week That Was. His most notable contribution was A Consumer's Guide To Religion, performed in the show by David Frost, which occasioned questions in the House of Commons.

Gillespie has directed many plays for the stage, most notably seventeen productions at the King's Head Theatre in Islington between the 1970s and mid-1980s, starting with The Love Songs of Martha Canary which starred Heather Sears. Tom Conti, Jack Shepherd, John Hurt, Tony Doyle, Nichola McAuliffe and Steve Harley starred in Gillespie's shows there. Notable productions were Spokesong, Tennessee Williams' Period of Adjustment, which Williams attended personally, and Punch critic Jeremy Kingston's Oedipus at the Crossroads, which starred Nicky Henson, Raymond Westwell and John Bott.

Film appearances include the Pride segment of The Magnificent Seven Deadly Sins (1971), The National Health (1973), Barry McKenzie Holds His Own (1974), Force Ten From Navarone (1978), The Thirty Nine Steps (1978), and the 1996 Royal Shakespeare Company production of A Midsummer Night's Dream. Robert recently appeared in Woody Harrelson's ambitious live-action movie Lost in London, playing the part of the mystic cabbie. He is working on Mike Leigh's new project, Peterloo.

Gillespie was the star of the Brian Cooke situation comedy Keep It in the Family, playing the harassed Cartoonist Dudley Rush, a part that Cooke wrote especially for him. The show ran for five seasons transmitted between 1980 and 1983. It also starred Pauline Yates, Stacy Dorning, Jenny Quayle and Sabina Franklyn.

As an actor, he performed in David Lan's Paradise at the Royal Court Theatre, John Arden's The Hero Rises Up at the Roundhouse, Peter Hall's Playhouse Theatre production of Tennessee Williams' The Rose Tattoo (starring Julie Walters); and in 1994 for two and a half years with the Royal Shakespeare Company in Twelfth Night, A Midsummer Night's Dream, The Broken Heart and Zenobia, which involved touring the US. He has worked several times with Adrian Jackson, founder of Cardboard Citizens, playing Luka in The Lower Depths and Charlie in Mincemeat.

Gillespie created his own production company, Jane Nightwork Productions, in 2000. Productions have included David Mamet's Oleanna, Jeremy Kingston's Making Dickie Happy, Deborah Cook's Sex, Death and a Baked Swan and Eugene Scribe's Golden Opportunities, translated by former Times Arts Editor Anthony Curtis, which received its UK premiere at the Warehouse Theatre in Croydon in September 2006. In May 2008 he directed a reading of Chains by Eugene Scribe at the Trafalgar Studios.

On 6 April 2010, Gillespie's new production, Love, Question Mark opened at the New Diorama Theatre for a 4-week run. Love, Question Mark is the first part of a trilogy entitled, Power of Three: Love, War and Death. The play starred Clare Cameron and Stuart Sessions and was produced by Lucy Jackson.

Gillespie has also written, directed and produced three plays comprising a trilogy, Power of Three: Love, War and Death. The first part, Love, Question Mark, was performed in 2011 and starred Clare Cameron and Stuart Sessions.

His autobiography, 'Are You Going to do That Little Jump', was published in 2017.