Depends on the POD but perhaps WW1 ended in a negotiated peace in which Britain lost face and a "stab in the back" was believed to have occurred.
I'd want a bit more specifics to flesh things out more, but I suppose that's workable. Not particularly likely, I should emphasize, but possible if the post-war years go badly enough. To answer your OP questions, the monarchy probably stays because fascism makes most of its allies among conservatives wanting to preserve the existing social order. And the transition wouldn't happen quickly. It would depend on continued malaise and disappointment in the realm of foreign policy especially, giving people the sense that the war was for nothing. Ireland would be an obvious potential sore point, and the Unionists in Ulster would be obvious allies for fascists in England.
As for their leader, so much time needs to pass from the POD that you could justify a lot of people. Moseley is kind of out of fashion for this role in timelines for being too obvious, and I think the time when it would be Winston has also passed. Apparently there was a trend of having it be George Orwell for a while, for maximum irony, and I saw T.E. Lawrence used in one timeline a while back. A woman would be unconventional, but Rotha Lintorn-Orman was one of the leaders of the British fascist movement IOTL, and armored warfare pioneer J.F.C. Fuller had some far-right connections AIUI. For ironic choices, I kind of want to see Lloyd-George in this role to reflect his mystifying support of Hitler in the 1930s IOTL.