It's then a question of other events and butterflies - the invasion of Russia is going to happen and in 1941 but does it happen early enough to be decisive or on 22/6/41 as in OTL? Bringing Russia into the conflict is going to fuel communist resistance everywhere (including the UK - why not?). The bigger question is or would be Pearl Harbour and the ramifications thereof. If BOTH Russia and America are in the conflict by early 1942 it becomes much harder for the Axis.
Trends - over time, German military control will strengthen over Britain as resistance increases and presumably the threat of direct American intervention via Ireland grows from say 44 onwards. As in France and Denmark, eventually the puppet Government will be so emasculated as to be meaningless and Britain will be under military control until that ends somehow - either a peaceful German withdrawal after a surrender following atomic bombs over central Europe or as a combined American, Canadian and Australian force lands on the south coast of England and the weakened German forces are pushed back.
After, the collaborators, both in Government and elsewhere, would be hunted down and tried (and executed) for their crimes. We'd see the same treatment meted out to girls and women who fraternised with German soldiers as we saw in Occupied Europe and the Channel Islands. As to the fate of Britain's Jewish population, well, that's a subject for another day.