So, the United Kingdom has surrendered to Nazi Germany, lets say late 1941. After Russia but before America enters the war.
No it was not sealion. But the British airfields and factories have been destroyed faster then they could be repaired and the luftwaffe control the skies.
Over all WHY they surrender isnt the most important part.
What happens after the surrender? Will German still declare war on the USA? With no longer having a Western Front at all will Germany be able to focus on the USSR and win in the east? What happens to the Europe under a Pax Germania? Who rules Britain after "independence" is granted? How long untill the resistance forces crumble? Will Fascist Britain go after Ireland? Will German bring Spain and Portugal into their sphere? Will Germany finish uniting the German speaking peoples with an invasion and conquest of Switzerland, what about the German speakers in North Italy? Finally, will a British Government in exile form in Canada or the USA and continue on the war?
You would have to have an actual occupation of the British Isles so Sea Lion has to work.
What part of the Royal Navy survives Sea Lion will head to Canada as will the King.
Why would Britian be "fascist" - it might have a fascist puppet installed but at worst it would be like Vichy.
Taking Ireland would not be left to the British - it's too easy an entry for Canada / Free UK / USA to re-enter Europe. It would be Nazi occupied.
I would suspect that Western Desert Force (whatever is left) would either be evacuated to India and then on to Canada via the Pacific or would evacuate to Kenya / South Africa - I think the first is more likely
India would probably be ungovernable in the short term - long term it either cuts a deal with Free UK for its independance or just declares it anyway. What Japan does is crucial here as a threatening Japan may persuade Free India to stay in the UK sphere.
Spain, Portugal and Switzerland are irrelevant provided they toe the German line - which they probably would. Germany would demand (and probably get) bases in the Azores and the Canary islands.
Free Uk would not accept defeat. Whether it could actively continue the war would depend alot on the Dominions and India. With them on its side it is still a significant power
Monroe doctrine may kick in to protect Canada and the Carribean. I'd expect this to be extended to Australia and New Zealand too. South Africa is probably most likley to jump ship and cut a deal with the Germans, particularly if it ends up with both Rhodesias. Italy will absorb Egypt and Sudan and will argue with Germany about Kenya