Always an interesting what if - could they etc etc.
So when you see the claims of the german fleet being 1 x BB [Scharnhorst]; HC [Hipper] & 2 x CL plus 7 destroyers and escorts, Thats the post Norway fleet [IE June 1940].. When the invasion was planned for the end of September and the beginning of October the situation was quite different.
they could rely on the following….
BB Scharnhorst [ 9 x 11” + 12 x 6” plus 14 dual purpose 4” flak and 26 flak & 3 A/C ]
BC Scheer [ 6 x 11” + 8 x 6” plus 8 dual purpose 4” flak and 16 flak & 2 A/C ]
HC Hipper [ 8 x 8” + 12 x dual purpose 4” flak and 46 flak & 3 A/C ]
3 x CL [each with 9 x 6” guns + 6 x 88mm plus 20 flak and 2 A/C ]
16x DD [each with 5 x 5-6” guns + 10-12 flak ]
8 x TB [each with 4 x 4” guns + 8 flak ]
43 x Corvette/Escorts [each with 2 dual purpose 4” guns + 4-8 flak plus sonar’s and 4 x depth charge]
13 x minelayers various converted ships with a compliment of 4-5” guns plus flak
~ 30 x heavy Sperrbrecher [armored mine breakers] with 2 dual purpose 4” flak and 17 light flak.
~ 50-60 tankers/freighters [ with 3 x 6” guns & 6 x Flak]
9 auxiliary cruisers [converted merchants with 6 x 6” guns & 6 flak plus several A/C]
There also looks like 1240 River barges [¼ powered by November and the rest towed] plus about 900 towing trawlers/tugboats and 700-800 steamers /freighters of various sizes.
The steamers and warships could sortie from German ports and meet up with the barge fleet escorted by the fleet of corvettes [know to the Brits as “Channel Destroyers”], for the amphibious invasion. The landing force would have firing support of 15 x 11” guns 8 x 8” and ~ 70-120 x 6” guns. The extent of the British coastal defenses in 1940 was slim indeed. While its believed that a string of British fortifications had been erected , these were mostly just bunkers that should fall to heavy coastal bombardment. The German Fleet flak amounts to ~ 200 heavy flak [88-105mm] and ~1500 light flak [20-37mm] which should provide adequate defense against RAF bombers that get through the Luftwaffe fighter cover to the armada.
The RN could count on 76 destroyers and destroyer escorts in the “anti invasion fleet”, mostly older destroyers [4-8 x 4” guns & 2-4 flak] plus a dozen CC [Heavy (8 x 8” + 4 x 4” & 8 flak) or Light (8-9 x 6” guns & 16 flak)]. With the 1/3 rotation , that leaves about 4 x CC and 25 x DD/DE to intercept the invasion fleet covered by 2 x BC , 4 x CC/CL & 24 DD/DE…looks like an even match.
The “ Home Fleet” could risk sorties through the “Channel”, with 8 x BB /BC plus 2 x CV and 15 x CC/CL escorted by up to 32 x DD , however most of these capital ships had only 8-16 flak, making them extremely vulnerable to Stuka dive bombing attacks. While its true that the Luftwaffe had little experience attacking warships , it didn't seem to have stopped them sink/damage ~ 1/4 of the UK fleet [~ 100 out of 400] sent to liberate the troops at Dunkirk.
This may force these capital ships to night attacks only , however they would remain always vulnerable to U-boat interceptions day or night. As with the anti invasion fleet, only 1/3 of these warships would actually be available for sorties. That’s up to 3 x BB/BC/CV ; 5 x CC/CL and 11 x DD. It may make sense for the Germans to sortie the Scharnhorst , the Scheer and Hipper to draw the “Home Fleet” away from the invasion, as it would take this entire force out of the channel action for some time.
Needless to say there are other RN warships are far a field, that could intervene but these could take weeks to complete their duties [convoys etc] and make their way to the channel. These include up to 40-45 destroyers on convoy duties in the Atlantic ocean, and another 40 in the Med. Further their should be 8 x BB/BC & 3 x CV plus 18 x CC/CL . Again as above only 1/3 of the above mentioned warships would actually be available for intervention….in other words about 30 destroyers, 6 cruisers , 1 x CV & 3 x BB/BC.