If the British get there before the Germans, the Germans will have a landing fleet left. They will surely use it.
I doubt they'd try to conquer Norway, as that would be lots of cost for little reward. On the other hand, Britain would also be extremely risky, but at least rewarding. The best for the Germans (militarily speaking) would be to wait until the war against France is over.
At the beginning of WWII, Britain was pretty vulnerable to ground attacks, and her best corps was in the desert. If the Germans manage to occupy the British with a few fake landings (Norway, Britain) they might actually be able to get to Britain.
British air superiority would be a big problem - but not bigger than waiting for D-Day, and the same as in Norway. Also, if just a single bridge head could be established and kept for a while, artillery could destroy a lot of infrastructure in a 50+ km radius, Flak could destroy a lot of British planes without loosing even more German planes, special ops could act all over Britain (surveillance, sabotage, espionage, infiltration, ambushes...), and so on.
Another problem for Germany is the British fleet. But actually, nothing would be better for Germany than being able to sink all those ships with coastal artillery, bombers, torpedo boats, subs, torpedoes from planes, and so on, right in the British Channel.
Once the Germans know where they can land without being discovered at once, they can easily establish new bridge heads in other places, so that British resistance can be diverted, British forces can be attacked from more than one side, and lost areas (very likely at the beginning) can be replaced.
If everything went well, the Germans could get the whole southern coast within a few weeks, which would weaken Britain so much that they'd have to give up similar to France or get conquered within a few more weeks.
I'd like to know if the US would declare war under such circumstances, and if the US declares war, whether the Russians might be tempted to use the opportunity (though rather unlikely at that time, with their experiences in Finland and with Japan in the back).
If the US declares war, the British wouldn't give up, so Germany would have to continue there. The US would probably go to Northern Ireland if the situation is really bleak for Britain, and, not letting the Irish choose completely by themselves, make them an ally. As I don't see any chance of Germany against the US, that would lead to a similar outcome as in OTL, but even faster, as the US is in the war earlier.
If the US doesn't declare war, WWII is soon over and Hitler can decide freely what to do next. I suppose, he'd consolidate his gains before thinking about attacking Russia, especially as there probably were extreme losses in GB. Tensions with the US might be reduced by giving Northern Ireland to Ireland, by ruling out any claims for Kanada and Australia, by leaving GB after dividing her into her 3 states and making them independent with just some reparations, by returning most of the occupied French territories to France in exchange for some colonies, and some more. Though such modesty might be unlikely for the Nazis. That scenario even has a 1941 map now - thanks to Thande for the 1939 map.
The war against the SU would probably still happen in 1941, if the US isn't in the war. But as Germany doesn't need to help Italy, no delays, and as GB is out of the war, she wont help the SU. Might make that war easier for Germany than it was in OTL.