It might be assumed now that when Bradman retired after the war everyone regarded him as the greatest-ever batsman. That was not the case.
The old caveat about his weakness on rain-damaged pitches persisted, especially in England. A typical assessment was that Bradman was the most efficient run-getter and supreme when conditions were good for batting, but that he was not the complete package. In 1951 Pelham Warner asked whether Grace had been superior. In 1962 England's leading cricket writer EW Swanton claimed that Grace, Hobbs and Bradman were equal first, all dominant in their own very different eras. John Arlott always maintained that Hobbs was the greatest batsman. So did leading statistician Roy Webber. A dwindling but vocal minority in Australia still placed Trumper first, though most supported their Don.
Uncovered pitches were still very much part of the game. In two consecutive five-Test series during the mid-1950s, Neil Harvey averaged 108 in the West Indies and 19 a year later in England. He actually batted pretty well in that wet English summer. It was all to do with the pitches.
Once uncovered pitches disappeared and faded from memory, people stopped talking about them. And stopped holding them against Bradman. He became generally accepted all over the world as number one. At least until Tendulkar neared retirement, when several Indian journalists, such as Suresh Menon, claimed Tendulkar was now the greatest-ever batsman and cricketer.
As mentioned previously, cricketing folk in the Caribbean have tended to rely on what they see for themselves rather than what someone else has written thousands of miles away. At the time of Bradman's retirement, West Indian opinion was that the greatest-ever batsman was George Headley. At that stage, aged 39, he had scored 4252 runs in the Caribbean at an average of 90.46. (His final average there was 86). He averaged 69 in England and, unlike Bradman, earned a good reputation on rain-affected wickets. When West Indies visited Australia in 1930-31, Ponsford was leading run-scorer in the Tests and headed the averages. Constantine said he was their most formidable opponent, not Bradman. There was no Bradman fever in the Caribbean.
It is difficult to say how long Headley retained his greatest-ever accolade in the region. Perhaps longer in Jamaica than in Barbados, Antigua or Trinidad.