Lloyd-Pack was born in Islington, London, the son of Ulrike Elizabeth (née Pulay, 1921-2000), an Austrian Jewish refugee who worked as a travel agent, and Charles Lloyd-Pack (1902-1983), who was also an actor. He attended Bedales School near Petersfield in Hampshire, where he achieved A Level passes in English, French and Latin. He subsequently trained at the Royal Academy of Dramatic Art (RADA), where he worked with actors including Kenneth Cranham and Richard Wilson.