After digging around for some articles from Marxism Today from the late 1950s, I re-discovered a wonderful resource that has fully readable online versions of British and American leftist publications from the mid-1900s. The four main ones that would be of interest to labour and leftist historians would be:
Labour Monthly – a long-running monthly journal edited by CPGB theoretician R. Palme Dutt. Officially it was not a CPGB publication, but unofficially, it carried the Party line set out by Dutt.
Marxism Today – established after the crisis of 1956 to enable ‘discussion’ between CPGB members, the journal was a pretty standard publication under the editorship of James Klugmann. When Martin Jacques took it over in 1977, it marked the beginning of the journal as a lightning rod for the ‘Eurocommunist’ wing of the Party.
Anglo-Soviet Journal – I don’t know much about this journal, but it seems to be a Soviet publication in English. It was advertised in the CPGB press throughout the 1940s and 1950s.
The Communist – This journal runs from the late 1920s (when the CPUSA was still officially the Workers Party of America) to the 1940s (before the Cold War begins).
So historians, start digging!