Cork was born the youngest of three boys in Newcastle-under-Lyme to Mary and Gerald Cork, both Catholics of West Country origin. His grandfather, Archibald Cork, played non-league football for Port Vale F.C. in the 1910s. His father worked as a financial consultant. He was educated at St. Joseph's College, Stoke-on-Trent, before continuing his education at Newcastle-under-Lyme College.

While playing for Staffordshire, Cork made his Youth ODI début for England Under-19's against New Zealand Under-19s in August 1989. Later in August he made his Youth Test début against the same opposition. He played six further Youth Tests to 1990 and five further Youth ODIs to 1990. Following strong performances for Derbyshire in the 1991 season, he earned himself a call up to the England A team for their 1992 tour of the West Indies, where he played in two first-class matches against the Windward Islands and West Indies A. Later in the year, following further strong performances for Derbyshire, he made his full international début in a One Day International against Pakistan at Old Trafford. He took one wicket in the match, that of Inzamam-ul-Haq. Cork played infrequently for England over the coming seasons, playing just two ODIs each in 1993 and 1994, against Australia and South Africa respectively. However, in May 1995 he played in three ODIs against the West Indies, taking 6 wickets at a bowling average of 21.81, with best figures 3/27. Later in the season he made his Test debut against the same opposition. In his maiden Test match he hit the first ball he received for 4 runs before going on to score 30 runs in England's first-innings, eventually being bowled by Courtney Walsh. In the second innings he scored 23 runs before being dismissed by Ian Bishop. His first Test wicket came in the West Indies' first innings when he dismissed Ian Bishop. More was to come in their second innings when from 124/3, Cork unleashed a bowling spell in which he took 7/43 – the best figures by an Englishman on Test debut – to help dismiss them for 223. This performance highlighted that at the time, Cork was the best all-rounder in England and also got his name on the Lord's Honours Boards. It also earned him the tag of "the new Botham" from the media. Two Tests later he became the 22nd man to take a Test cricket hat-trick, when he dismissed Richie Richardson, Junior Murray and Carl Hooper in successive balls in the West Indies second-innings;