It would have to be a very tense, frightening, extreme political situation. I guess we have Oliver Cromwell as the closest thing to a modern British dictator? Though I'm not too sure about that.
Alright, so as this is a challenge I'd better lay out some specific ideas.
Firstly, the Royal Family have got to go. No no, I don't mean die. Just become deposed from their ceremonial roles. Why? They are, above all, a symbol of constitutional government and they still have - theoretically - some powers to use in emergencies. A dictatorship isn't going anywhere without monarchical support unless they are sidelined like in Japan, or perhaps a Mussolini-Victor Emmanuel type situation. I can imagine something of that sort.
Let's say the Royal Family are, in a revenge for the death of Hitler's close friend or something, viciously bombed and Buckingham Palace is brought to ruination. Nobody is willing to take the throne after the death of the King, and after an extended crisis Churchill takes a title (not sure what) and basically assumes monarchical roles - or at least that of head of state.
The war continues as normal, but as absolute power corrupts absolutely Churchill decides elections in '45 are going to lead to the decline of the British Empire - his most favorite thing of all - and suspends elections. There are large Labour protests immediately, but a swift crackdown and a message from Churchill promising democracy once the war is over eventually quells resistance. Churchill, more paranoid about losing control (and thus, in his mind, India and the Empire) beefs up security forces loyal to him, and decides not to "assist" in the choosing of a new monarch. (Yes, I know about the line of succession.. but let's say they've all had enough and just want to grieve, or are silenced by the government).
Outrage by Roosevelt is met by a retaliatory pact between Churchill, De Gaulle and Stalin - all leaders, at this stage, one degree or another not totally averse to somewhat undemocratic means. Hardship brought on by a lack of US aid due to consistent British imperialism leads to tough economic times but a powerful propaganda regime and constant defiance and speeches from the PM (and protector of the empire?) Churchill settle most hearts and minds.
The dictatorship is rather non-extreme by dictatorial standards. There is the occasional illegal execution of this or that communist spy - Stalin being willing to overlook this for his alliance of convenience, or rather more a "support group" in using UK and France against the US as a buffer as Stalin was wont to use nations as. Churchill hates communism but is willing to tolerate Stalin, as uneasy as he feels due to the fear of losing the Empire. The prisons fill up a bit, but on the whole the dictatorship can best be compared to that of the Argentinian Junta of the 1970's and 1980's, save the military government aspect. Freedom of speech is quite curtailed. A mouthing off of the DoE can lead to a visit from the police, and political activity not in support of whatever the Tory party now looks like is met with jail or at the least fines. As for the military, Churchill's patriotism and large budget blessings combined with skilled appointments by his cabinet ministers results in a mostly loyal and sympathetic armed forces, especially so given their new role in the governance of the country.
India is a lost cause to any sensible Briton - in the words of Gandhi (taken into hiding by Indian civilians, to the personal outrage of the DoE) how can 100,000 rule over 300,000,000? However Churchill is not really a practical thinker, and decides to increase the number of colonists in India by a million over five years from all around the Anglo parts of the Empire, prompting outrage inside of India. Guerilla activity and political movements are both harshly suppressed, much more harshly than in the home isles and extrajudicial killings are frequent against "troublemakers" and a few of the innocent who are caught up.
When the Indian War of Independence breaks out in 1947, Churchill relies on "divide and conquer" to win, warning Sikh and Muslim populations of the impending massacres they face in order to win support. Of course, despite attempting to disarm them the Indian Army is utterly overpowering and quickly seizes control of the major cities. A large British army arriving in Bombay a month later, backed by heavy bombers from Pakistan, begins a campaign to regain control, but progress is slow and eventually Churchill is kicked out by his own cabinet - men who tolerated him until now, just as the US decides to apply heavy sanctions to the new UK government - and whom Churchill hand picked for their loyalty. A quick peace and a recognition of independence to India is signed by the new cabinet as they begin to undo political restrictions - albeit slowly. About 45,000 Britons are killed and around 400,000 Hindu Indians and 350,000 Muslims (most from Hindu-Muslim violence) during the civil war. Of those numbers, the amount of combat troops killed is: United Kingdom, 25,000. Indian army, 80,000. Muslim militias, 70,000. Hindu militias, 40,000. Dismayed, depressed and rejected by the country he loves, Churchill is not tried by the still somewhat sympathetic cabinet as they try to decide whether a new monarch or a presidential republic is best for the future but rather left to his own devices. Depression overtakes him as he feels the scorn and anger of the public and sees the Empire he loved fall to pieces before his eyes, and upon hearing of the Sudan, that country which he had fought and seen men die in so long ago, becoming independent on 15 March 1950 (the Empire being far too economically weak to cobble itself together in any real way at this stage) he falls into an irrevocable pit of despair. Two days later he hangs himself with his leather belt.
The end.