Just a short reply WRT naval forces:
1.) The forces near the invasion zone - which were light forces, destroyers and a handful of light cruisers - would be among the priority targets for the Luftwaffe if the necessary air superiority had been established. And that is before the "invasion fleet" raises anchors.
Besides there is more than one way to take warships out of the equation. Damage them, damage or destroy the infrastructure they rely on (piers, cranes, ammo and fuel depots) and they won't be more than a heap of metal. Had the Luftwaffe won air superiority the big harbors of Southern England would have seen very heavy attacks (i.e. Portsmouth or Southampton).
2.) The Royal Navy was stretched thin across the globe due to the necessities of the Empire, the italian navy in the Mediterranean Sea and the need to escort convois across the Atlantic. The Home Fleet was based at Scapa Flow and a number of cruiser and destroyer flotillas along the Eastern Coast. Getting them to the area of operations does take time, a group of heavy ships coming from Scapa will take up to a day to reach the combat zone. Which means there will be no heavy cruisers or battleships opposing the initial landings.
Additionally the Channel was a narrow theater, rather easily controlled with light forces, recon planes, submarines and - the real trump card the Kriegsmarine had - mines. Large ships are best suited for large sea areas. They need space to maneuver ... which was just not there in the Channel.
3.) Battleships and the likes are political weapons. Regardless of their firepower, their protection or their speed there was always a political element in their use. Wilhelm II left the High Seas Fleet in their harbors since he feared losses more. The Japanese left Musashi and Yamato at home until it was too late for them to make a useful contribution. I think the british government would think hard and long about employing its valuable battleships and battlecruisers in these narrow waters, where the risk of losing them is very high while the potential gains are at least questionable. These ships represent a nation's prestige in naval affairs, losing them for nothing would be a serious blow.
I'm not saying a potential invasion would have worked for sure. I am saying we're talking about very muddy waters here and it's simply impossible to tell what exactly could have happened and how that could have effected other decisions.
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