The phrase appears to have entered English print around the time of West Indies' 1933 tour, after Headley made 169 not out at Old Trafford. The Manchester Guardian and a handful of Fleet Street papers used it casually — sometimes as headline shorthand, sometimes as compliment. Bradman, the implied reference point, made the comparison flattering on its face.
In Caribbean papers and in the Trinidadian intellectual circles around CLR James, the phrase landed differently. James, in a series of newspaper columns in the 1930s and later in Beyond a Boundary (1963), argued that the formulation was diminishing: it ranked Headley relative to a white Australian rather than admitting his independent claim to be the best batter in the world. James preferred 'the white Headley,' and recorded Headley telling him in conversation that he tried 'not to mind.'
Headley's Test average of 60.83 across 22 Tests was, until World War II, second only to Bradman's; in away Tests in England his average was higher than at home. His four Test centuries before the age of 23 matched Bradman's pace. He carried the West Indies batting almost single-handed across the 1930s; Wisden 1934 wrote that 'on a difficult wicket, his judgement is so cool that he would have to be ranked among the great batsmen of any time.'
The nickname survived into the 1950s, occasionally even applied (against the player's clear wishes) to Everton Weekes and Frank Worrell. It dropped out of mainstream English cricket vocabulary in the 1960s. Modern accounts of Headley generally either avoid it or place it in scare quotes.