It's extremely likely that an early to mid twentieth centuary comunist controlled Britain would be called Great Britain, especially given that the largest communist party was the Communist Party of Great Britain.
I think Great Britain is the most plausible name, after WWII when the ex-servicemen's families and squatters movement, and the National Union of Unemployed Workers under the influence of the CPGB, rose up against Churchil's 1946 re-election and established the 'New Unity Government', with the Labour Party and the TUC - this being brought about with the support of a section of big business as the NUG was the only force able to diffuse the general strike and the housing riots and bring back social peace, with the help of a very sympathetic armed forces. This has lead to the uneasy 50 year rule of the NUG and the excesses of the Scargillite regime in the eighties which brutally crushed the petit bourgeious small business men and free marketeers uprising of Margaret Thatcher.
Thank god for the limited reforms the technocratic current General Secretary, and Prime Minister Peter Mandelson, he may not be perfect, but at least we can choose more of what we spend our wages on, and even choose to get paid in a hard currency like Gelders now.