I disagree that we (Britain) would have been invaded by Germany had it not been for USA: maybe we would not have been able to liberate Europe but I'm sure we would have been able to repel any invasion and the Germans would have just given up. The Royal Navy and the RAF were extremely powerful and second only to the USA.
This is an interesting topic, so perhaps I'll start here.
German mistakes with regards to invading Britain commenced pretty much from the Invasion of France, and certainly from the point of Allied evacuation at Dunkirk. The best German fighter (Bf109) lacked range and was deployed from airfields which limited their operational time over the battlefield. British Spitfires and Hurricanes could operate from Southern England, and spend more time over the battlefield. Theoretical numerical advantage didn't mean much in this scenario. Similarly, German bombers (operating largely from Germany still) were at their range limit whilst carrying payload, operated in daylight, and with insufficient air cover due to the lack of fighter range.
German aircraft losses were quite heavy, despite the momentous German victory.
German policy at this point was to pursue peace, not actually invade England. Britain had a well-storied history of appeasement, and Hitler felt like peace would allow consolidation of his victory (in that he was quite right) and that he could extract good terms from a beaten down Britain (in that he was quite wrong).
British command had changed substantially, and was largely taking on Churchill's persona, as opposed to Chamberlains. The British sinking of the French fleet was a pretty clear indication of that, I think.
Contrary to popular opinion, a direct sea-based attack on England (Operation Sea Lion) was just one of a number of next steps identified by the OKW (German strategic planning). Where direct action against Britain was considered, air superiority both in the moment, and to restrict British aircraft production in the future was seen as vital to ALL identified approaches. Alfred Jodl, from the OKW, actually believed that combined air and sea operations could break the English spirit. This mistaken belief was not unique to the Germans, it is worth noting.
In attacking Britain, the Germans decided to attack the RAF directly, along with it's support mechanisms and ground forces. This would then lead to effective attacks on production lines, imports, etc. Interestingly, they didn't separate Fighter Command from Bomber Command...they simply saw the entire RAF as the enemy. Doesn't really make sense to me, and it had consequences...if the intent of attacking the RAF was to destroy their ability to defend production and imports, then Bomber command was really quite beside the point.
The same range issues of the Bf109 impacted here, too. German fighters could only operate over Southern England. Whilst they never had to, the Brits were ready to withdraw Fighter Command to the north, outside of German fighter range. And production centres in Liverpool (for example) were never really impacted.
Initial German intelligence was way off the mark, too. It underestimated Hurricane and Spitfire numbers, had no mention of radar-controlled air defence the RAF relied on, and believed the Luftwaffe could win the air battle over Southern England in...wait for it...4 days. This would be followed by 4 weeks of bombing to destroy the aircraft industry. So...1 month for the destruction of the RAF, and the destruction of the British aircraft industry. It wasn't just wrong...it was delusional. This is partly the problem when you have a leadership structure only interested in good news.
The Germans also had a further complication. Where a fighter was shot down, the pilot could...sometimes...eject. German pilots ejecting over England, or the Channel had far less chance of becoming operational and available crews again than English pilots. German aircraft production was problematic, as was some basic logistical supplies (for example, despite using drop tanks for Bf109s in Spain, none were available in 1940...!). But even where they HAD sufficient of their best aircraft available, they lacked crews. And certainly, they lacked experienced crews.
Despite ALL that, the RAF was pretty close to breaking at some points. German attacks on radar installations were always haphazard and ineffectual, but it wouldn't have taken much more focus to have really damaged the umbrella covering the coast. And the Brits ability to shift their resources as required was a major factor in them being able to use their smaller air-force effectively.
But it wasn't sustainable. German aircraft production was basicallly still on a peacetime footing. They lost 1636 aircraft between July and Sept 1940 alone. That's over a third of the number of aircraft they commenced with. Ultimately, when the initial German assessments proved horribly wrong (ie. the Brits couldn't be broken by aircraft and/or submarines) they had no plan B.
One of my favourite lines (translated to English) was from the Italian Foreign Minister (Galeazzo Ciano). He suggested Hitler was like a gambler who had made a big scoop, and would like to get up from the table, risking nothing more.
Of course, England didn't sue for peace, and Germany had no clear plan B.
With 20/20 hindsight, I think I could tweak German strategy to a point where the Brits just might sue for peace. But this would involve a lot of work in isolating Britain from allies, whilst strangling British supplies...not tit for tat daylight bombing of London, alongside unrestricted submarine warfare which sank American ships. And the point would NEVER to have been an invasion, but instead to get favourable peace terms.
Once the Germans turned to Russia, their designs on England were done. They simply couldn't focus on a multi-front war.
I'm over-simplifying, but trying not to write a book...!