I believe a widespread criticism of Mosley, voiced by critics as varied as Hitler and Orwell, was that his obsession with militaristic aesthetics was alien to the British character. I wonder how this would would manifest itself in an ATL of a CP victory.
So there's this great lil kaiserreich after action report called the Crown Atomic. Its not super plausible in terms of its initial premise (an exiled British government turning the Empire effectively fascist to reclaim the British isles ironically from a Stalinist Mosley) but its presentation of British authoritarianism I think is probably pretty spot on. It basically revolved around an alt Edward as a King-Emperor at the centre of an imperial cult of personality, who is portrayed as this ageless figure with a carefully manicured public persona. Propaganda promotes "a glorious past" and a bold brave future, the idea being to create a pan nationalist ideology that worships the Empire as a collective enterprise that was "almost lost" and must be safeguarded no matter the cost. This is seated in collective and desperate misery from two world wars and oceans of spilled blood. The population of the Empire in this alt is scarred by decades of war and grief, with sacrifice and martyrdom of fallen being core.
Parliament and democracy technically still exists but both are controlled by a technocratic shadow elite of aristocrats, industrialists, business magnates and military men answering to the King. This elite promotes an illusion of democracy but in reality, all political parties subscribe to the same ideological consensus and non compliant parties are not allowed to exist. The economy is state controlled and social credit is a tool used to control the populace, with service to the state as a soldier or bureaucrat lauded above all else. Consumerism is encouraged to basically anesthetize the population to the erosion of Democratic and civil liberties, and after decades of instability, a lot of people subscribe to this.
Limited workers rights are encouraged, although these are stripped away at the slightest hint of non compliance and are a tool of control rather than being a genuine concern for worker well being.
Women are encouraged to work and even serve in military auxiliaries, though cynically this is to make up manpower shortages the Empire faces, and things like prophylactics are state controlled to encourage women to also have children.
Children are mostly subject to propaganda at a young age and they are taught about the virtue of the Empire (encoraged to view it as a giant family under siege). Talented children are taken to join "pioneer schools" which brainwash them into to serve the a Empire and incorporate them into the technocratic elite.
The Empire also cynically embraces multiculturalism, though this is surface deep at best. Being an Empire, keeping the likes of India and Africa under thumb is important, and this manifests with the Empire paying lip service to Indian/African nationalism, if only to keep them placated while they exploit the populace and resources of both. Africans, Indians etc can advance in society, but there is a low ceiling for most.
Virtually any leftist sympathies, real or imagined, leaves individuals sidelined by society at best or imprisoned/dead at worst. Penal colonies in Australia and Canada exist for those that speak against the Empire, secret police round up and torture dissidents and dobbing in neighbors or family is encouraged.
Basically everything is a warped version of things Britons (at the time) cherished.
Militarism is promoted as duty, the monarchy is overblown and the Empire is the centre of this "glorious" nationalist mythos.