Born in Newcastle, Barnwell first played as an amateur for Whitley Bay and Bishop Auckland before moving to Arsenal in 1955; turning professional in November 1956.
A semi-regular in the Arsenal side for the next six seasons, Barnwell initially played as an inside forward, behind the main strikers, but during 1961–62 he was switched to a more defensive wing half position.
However, Barnwell returned in 1962-63, playing 37 games at wing half, only to lose form again the following season, however, and he was dropped from the side.
However, Wolves could not maintain their form at the turn of the decade, finishing 18th in 1980-81, although they did reach the FA Cup semi-finals for the second time in three seasons.
He then returned to England in the lower divisions to manage Notts County (1987–1988), Walsall (1989–1990) and Northampton Town (1993–1994).