What if Germany won the Battle of Britain and tried to invade

Much has been said about the ominous photos of invasion
craft assembled for operation Sea Lion.
I wonder the the Germans meant these as pure intimidation
British intelligence must have found them almost laughable, horse drawn canal craft
rounded up from all over Europe, meant to pulled across the channel by tugs.
Only a few were powered.
　
This quote from the Imperial War Museum book of The War at Sea, puts it thusly,
"The chaos that would have ensued once these barges were released to proceed to the beach, without an adequate source of propulsion, can be imagined..........they required 4500 horses...German amphibious techniques were hardly better than Napoleon's."

It makes me wonder if the British need have worried at all.

 

The British military ran a Wargame in the 70s to see what would have happen.

Basically Germany managed to get a beachhead but failed to keep the supply lines open across the channel and the operation failed.
Likewise any English attempt to establish a beach in France would have meet similar results. Which is why Hitler was confident to right off England and move towards Russia.

It took D-day to break the Atlantic Wall. D-day showcased that an ungodly huge amount of resources was needed to establish a cross channel beachhead and keep it supplied - and even that a high risk of failure. For Germany to attempt essential the equivalent of D-day in reverse with a lacking Navy would have been a huge gamble.

Which is why many German Generals at the end of the war said it was a bluff. Hitler was trying to spook England into negotiating. That he didn't really intend to go through with Sea Lion. Germany felt that they could defeat Britain in other ways anyway. And even if they couldn't, like I said it was mutual. England couldn't really invade Europe either (at the time Hitler didn't predict the massive military power to be of the United States).

But your what if is if the Battle of Britain was won by Germany. That is a large statement. Best case scenario for Germany, they suffer little casualties and dominate the RAF. Then they still probably wouldn't invade, because of the Navy issue, but rather just continue to bomb England until she surrendered. A better question would be what would happen if Germany had the French Navy - that is on another topic.

 

In order to answer the question you really need to better define what you mean by the LW won the BoB. For instance the most likly "win" for the LW is that the battle continues as it was a bit longer and the RAF decides to pull 12 group back from the south. This means that the LW is also highly attrited and when the invasion comes the RAF may actually be significantly stronger than the LW and since they were being held back to await the invasion things don't work out well for the Germans. Then there's the fact that in the case above the invasion would likely be pushed into October when weather becomes very questionable.