- If the Axis forces had air and sea superiority would the invasion succeed? - very possible
- Would partial air and sea superiority be enough to force a landing? - yes, but it wouldn´t have lasted
- Did Germany have the logistics and supply lines to fight an ongoing battle in Great Britain? - if the MBL/defensive paramters were established it would have been very likely
- What shape was British armed forces in after the BEF retreated from France? - similar as all other beaten armies of WW2 - fanatic or fatalistic or both
- How would it have shaped WW2? - a great deal to the favour of the axis
Reasons
to 1.) - To this time no one had an real answer for the Blitzkrieg tactics. If both, air and naval superiority, were achieved Great britain would have been cut from all supplies and reinforcements. The Germans had had the opportunity to send over their forces without a real threat, assembled them and started a well prepared strike. Much of the heavy military material of the BEF was lost in France. Further Tauchpanzer (Diving Armor) were tested and made combat ready on axis side. To the times of WW2 an absolute novelty. I think in reality they were used to cross the Don in russia after the Sealion OP was cancelled and showed their value. The russian T series of the postwar period were in general built after that principle.
to 2.) - A landing could have been forced but would not have lasted because of lacking of secure reinforcements and supplies. And what would have been much more important the costs of material and men for such a landing would be too high to have more as propaganda purposes. In the end the landed forces would have been cut and outworn/battered so much they would have lost their use.
to 3.) - Supposed the landing was a success and the superiority to sea and air established they would have had enough logistics because no military secured transports and armored ships were needed anymore. They could have used the much-maligned river barges and ships. One should remember the transport space of them would have been enough. Only the security of that barges and ships against enemy fire and rough sea etc. was too weak or not even existent. But if one has the control it is almost as shipping in peace (metaphorically spoken). A fuel pipeline would have been erected on the ground of the canal. Me-321 and similar types would have transported soldiers en masse or tanks and guns - and that added to the naval supply. So yes, they would have had the logistics.
to 4.) - In reality both, fanatism and fatalism, is a real problem in fighting. An opponent who is one of this is unpredictable. And that is the worst what could happen. A fanatic is ready to sacrifice all included his own life to gain his goals. That is what makes suicide attackers so dangerous. For a fatalist it is of no matter anymore what is happening so he follows his learned patterns or tries to sell himself well. That, too, makes a man dangerous. For an opponent who has to fight one of that both, fanatic or fatalist, it makes no great difference. The result is most of the time one has to fight someone who goes till to the end and surrender is no option for him.
So I think it would have been for the Brits as it was for the Germans as the Russians pushed them back to Berlin. The losses on attackers side would have been very high but in the end the defender had lost.
to 5.) - One has only to imagine the possibilities. The Germans would have learned by doing what it means to prepare and execute such a naval invasion. They would have been prepared after that against an allied naval invasion - the west wall would have gotten a higher priority. Further the allied naval invasion in the Normandy would have been cancelled. No multiple fronts war would have been possible anymore. British colonies would have been broken over the time - at least the possibility would have been risen. Maybe even civil war there. The german forces had had a huge base for aircrafts, ships, submarines etc. what was not encircled on all sides by enemy nations etc. etc.
I don´t know if that would have been enough to win the war. But in every case the constellation for the end of the war would have been changed dramatically.
Greetings