Maybe if Communism had been more successful elsewhere it might improve the prospects of a Communist revolution in the UK? Say the Reds win in Hungary, Finland, and maybe the Baltic states. Austria also came pretty close at one point to Communism as I understand it. As a result the Soviet Union is in a much stronger position economically and militarily, which in turn mean both stronger appeal for revolutionary socialism, as well as stronger opposition to it.
Then prevent the worst excesses of Soviet degeneration. Perhaps the Left Opposition is able to win out against Stalin, or they form a popular front with the Right opposition against him? Maybe even have Makhno and the anarchists incorporated into the Soviet project, rather than violently suppressed. This doesn't have to lead to a perfect socialist democracy, but it would take some of the sting out of anti-Soviet propaganda. Also if Trotsky and the Left Opposition remain significant players in Soviet politics that means no 4th International, which butterflies away a major source of far-left sectarianism.
In response to a more active and aggressive far-left, the far-right gains more mainstream support. The 1926 general strike goes badly, leading to the government declaring martial law and establishing a semi-military dictatorship. The military government in Spain is less willing to surrender power, which leads to an early Spanish Civil War (perhaps the 1934 Asturian miner's strike escalates to a full uprising), with Communist/Anarchists as a significant faction, and the UK supporting the military government. With the depression setting in, rising demand for a return to democratic rule, and opposition to UK involvement in the Spanish Civil War growing, all of which contribute to an increasingly radicalised working class and trade union movement, and with the government relying more and more on violent repression and fascist thugs to maintain order, the stage is set for a very British revolution.