Unless the OP is mistaken, the professor is full of it. Kershaw generally keeps mum about his personal politics, but to the best of my knowledge, he was not a member of the Communist Party of Great Britain (CPGB). There was a large cohort of British historians active in the CPGB like Eric Hobsbawm or E. P. Thompson, and Kershaw's name never comes up in discussions on them.

Additionally, applying the label "communist" in the context of the British academy really does not tell you that much. The CPGB was a somewhat peculiar institution and many of the historians in its ranks were apostates from the conventional Marxist-Leninism of the Party. Members of the Communist Party Historians' Group like Thompson and Hill broke with Moscow over the 1956 Soviet invasion of Hungary. Their identification with communism was much more anti-establishment and trying to parse out a history from below. Politically, Hobsbawm and his compatriots tended towards Eurocommunism, which was a much less doctrinaire and non-aligned form of Marxism that was at odds with the USSR and some of the CPGB's leadership.