Bancroft was born into a musical family environment that moved to Scotland when he was nine years old. Playing saxophone, he joined a trio led by his pianist father and completed by twin brother Tom Bancroft on drums. During his first outings as sideman and leader, tenor saxophonist Bancroft was sometimes heard in the intriguing mix of jazz and contemporary Scottish folk music that so enlivened the UK jazz scene from the mid-90s onwards. Among musicians with whom Bancroft has worked are guitarist Mike Walker, bass players Reid Anderson and Steve Watts, and drummers Thomas Strønen, Marcello Pellitteri and John Rae. With the latter he was a founder member and musical director of the John Rae Collective and also joined him in Celtic Feet, along with Brian Kellock, bass player Mario Caribé, concertina player Simon Thoumire and violinist Eilidh Shaw. Bancroft also played with Sun Ra, Tommy Smith, and Kenny Wheeler. In 1994, the Bancroft brothers and guitarist Kevin Mackenzie formed Trio AAB, a group that toured and recorded the albums Cold Fusion and Wherever I Lay My Home That’s My Hat achieving considerable critical and popular success. For their third recording, Stranger Things Happen At C, the group added flautist Brian Finnegan and demonstrated that this particular vein of musical thought remains rich. By the early 00s, Bancroft had proved himself to be one of the most imaginative musicians in the UK and when he moved into a more impressionistic mode with his second solo recording, Headlong (2004), he again attracted rave reviews.