Bradshaw was born in St. Paul's Village, St. Kitts, on September 16, 1916, to Mary Jane Francis and William Bradshaw. His parents were among the few in the community w'ho did not work in the sugar industry. Mary Jane was a skilled cook who was much in demand and his father was a blacksmith who abandoned his family nine months after his son's birth. Bradshaw grew up within a close-knit family that included his grandmother, aunts, and uncles, who had high expectations for him and provided support. While still in St. Paul's Anglican primary school, he distinguished himself for his academic and leadership skills.