Conway was born in Bristol, England. He had no formal piano training and spent his early adulthood in the Navy. Conway was talent-spotted while playing in a London club, signed to Columbia Records and spent the mid 1950s providing backing for their artists including Gracie Fields and Joan Regan. However, the commercial potential of Conway's unique percussive piano style (which he attributed to a missing fingertip he had severed many years previously with a bread slicer) was too great for him to remain in the background for long.

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