I remember when I was young, so quite a while ago, reading a book based on a wargame that some experts ran not long after the war. They got people in to play the main characters, e.g. Adolf Galland played Goering, etc, and they followed what were known at the time to be the invasion plan and the counters to it! And of course they knew what the weather had been like.
Basically if the Germans had invaded without winning air superiority. The battle would have been a mess on the ground, but the Germans would not have been defeated until the entire Home Seas fleet had sailed into the Channel, destroyed the invasion fleet and most of Calais. The entire fleet was lost but they had done their job!
With complete air superiority, the Luftwaffe would have bombed the fleet as it sailed south, so it wouldn't have got anywhere near the Channel. Without air superiority, the RAF could protect the fleet until it was in the Channel.
The turning point was caused because not losing the BoB, allowed Roosevelt to win his arguement to support us, and not Joe Kennedy's, to abandon us!!