Bradshaw was something of a late bloomer, making his League of Ireland debut at the age of 26, before then going onto enjoy a brief but meteoric career during the seasons approaching the Second World War.
Bradshaw was born in a poor part of Dublin and left school at an early age to help supplement his family income, working mainly on the docks.
He played for an Ireland Junior team against a Birmingham FA XI at Dalymount Park and scored a hat-trick in a 4–3 defeat.
Bradshaw opened the scoring after just 20 seconds, the quickest ever goal in the history of the FAI XI, when he blocked an attempted clearance by the Swiss goalkeeper, Huber.
On 13 November 1938 Bradshaw won his second international cap as he helped Ireland to a 3–2 win against Poland, again at Dalymount Park.
Playing in a team that also included Jimmy Dunne, Kevin O'Flanagan and Mick Hoy, Bradshaw scored in both games.