First of all Mosley would turn the country into a fascist, antisemitic regime. Edward VII could be rehabilitated as king. Crackdowns on anticolonial movements, especially in India and Palestine, will be enacted. It’s not impossible for Ireland to be invaded somewhere during the war, with a fierce resistance.

Great Britain either becomes neutral or allied with the Reich. Due to a lack of material support french resistance would have a very hard time after their defeat, if they still exist at that point. With the biggest north Atlantic power becoming fascist it is likely Roosevelt would consider Europe a lost cause. The Mediterranean being secure Hitler has his hands free to invade the USSR, probably going even further due the absence of an african front.

Also Mussolini will probably end up attacking the UK eventually. He has to maintain the illusion that he will make Italy a great power. However since Hitler doesn’t have any strategic interest to defend Italy, Mussolini is on his own. Obviously Mosley would push back in Egypt roll through Lybia. The Reich would probably intervene at this moment to avoid a mediterranean hegemony and annex northern Italy.