Basically the Germans succeeded at first but had heavy casualties, the defences worked in their role of slowing the Germans down. However, by the second day, the Royal Navy had come down from Scapa Flow and dealt with the minefields and other protective measures put down by the Kriegsmarine so they were causing havoc with the german supplies crossing the channel.
Eventually, the Germans were running out of reinforcements and supplies, whereas the good guys were bringing down more reinforcements and supplies from the north, eventually German commander issued a withdrawal. Would have been very heavy casualties on both sides, and South East England would sadly have suffered massive civilian casualties, especially the home guard as the Germans treated them as spies pretty much as they weren't considered legal combatants.
Fascinating stuff, would make a great movie, even more useful for me as i'm currently studying the Battle of Britain for my history A level coursework.