When this was gamed in the 1970's at the Army Staff College at Sandhurst, the result was that although the Germans would get ashore - they wouldn't be able to extend at depth into Kent and Suffolk. After three days the RN Home Fleet in the North Sea swept aside the weak KM forces in the North Sea and entered the Channel - breaking the supply/reinforcement "bridge" and bombarding the bridgehead from the sea.
Against the factor that IF an invasion was launched, by Hitler's directive it would only be under the umbrella of LW air
superiority - note, NOT supremacy
- and the military spremacy of the Whermacht against the evacuated and reforming British Army - a number of other factors have to be set.
First of all - the Royal Navy. Despite loosing a large number of SMALL ships at Dunkirk, the RN still could field over a hundred destroyers around the world - and formed an anti-invasion force of 36 of them, with three command cruisers, in Portsmouth, Southampton and the Medway. Against this, apart from the KM's u-boat forces - as of the summer of 1940 HALF the Kreigsmarine was sunk or under repair after the Norwegian campaign. I think the numbers were only FOUR capital ships (cruiser weight or above) and four destroyers undamaged and useable after the capitulation of Norway.
Now - it's often said that the LW could prevent the RN operating in the Channel against the Sealion fleet; the lesson of Norway when the
Warspite's task force was attacked by the LW, and a year later off Crete - was that although a surface navy WAS vulnerable (of course, like ANY target) to air attack...a
balanced force of small ships, unladen by the tons'-weight of Dunkirk or Crete evacuees. The
Warspite was damaged, and ONE destroyer sunk in
TWO days of prolonged attack off Norway. Off Crete, the RN took losses during the evacuation because two of the lost ships had to put to sea again without having a chance to restock with AA munitions. Fully stocked - and the English Channel was definitely within reach of resupply - RN destroyers and AA cruisers COULD stand a good chance of holding off air attack.
Meanwhile - and a good source for the next part is Peter Fleming's book - the invasion itself would require around three weeks' of prolonged sailing by the invasion flotilla back and forth across the Channel. EVERYTHING would have to be moved in an environment with no Mulberry and very possibly no UK ports being taken until the second or third week. To all the human materiel required to be moved - hundreds of thousands of tons of HORSE FODDER would even have to be moved by the fleet to keep the Wehrmacht's "transport" moving
That's THREE WEEKS at a minimum that the Germans have to keep enemy air or sea assets from breaking the Channel "bridge"....
Against this you can now add the efforts by night ( and day) of Bomber Command historically in bargebusting, literally night after night through the summer and early autumn of 1940. As well as facing repeated demands for the barges gathered, the backbone of Europe's riverine traffic, to be returned to keep Europe's raw material moving, they eventually had to be moved further and further away from the shortest crossing point and original mustering points in an attempt to reduce the RAF's depredations.
I've yet to see a good study in ONE LOCATION of who fast the British Army actually DID rebuild and re-form after Dunkirk; one thing that is often neglected however is the 225,000 men of BEF II that were evacuated in June from Brittany and further south; THESE men didn't arrive back in the UK in the same state as the original BEF in Flanders, they only arrived back
sans their equipment, they had suffered VERY little in the way of disorganisation or casualties or other losses - having only been landed in Atlantic-facing France AFTER the BEF began to break in Belgium! The only major loss to BEF II was the 51st (Highland) Division which surrendered at St Valery-en-Caux.
In fact - if you look closely - ALL the comments by Prime Ministers, generals etc. about the disorganised state of the British Army and its lack of equipment ALL date from
June 1940; very soon, in only a matter of weeks, comments like "the Canadian 1 st Infantry Division being the only combat-ready formation in the UK"
soon vanish from the historical record
People often point to the Home Guard and say the British were bound to be beaten if that was all they had to stop the Wehrmacht; they rarely look at the whole view of the prospective battlefield...what the British were planning to do was what caused the US tens of thousands of casualties on Okinawa and made them paranoid about invading the Japanese Home Islands - on land they intended to fight thir MAIN battle
inland - away from the coast. British coastal defences - and may of them can still be seen today alongside various stretches of the coast, and seen in pictures of the period - were remarkably weak compared to, say, the later Atlantic Wall. The British visibly didn't intend to make their stand on them, they were JUST to delay the invaders. Likewise,
the Home Guard wasn't ever expected to defeat the Germans - just sponge them up, cause them casualties and delays. The one thing that France HAD cost the British Army was its
[armour - and though the Army's complement of 25pdrs could be back on establishment by August, replacing tanks was far more difficult for it depended on the output of steel foundruies, armourers, engine companies, AND the factories cobbling the whole thing together at the end!...
so quite simply the British Army intended to fight an "old-fashioned" battle of artillery-stiffened static defence lines and do without its armour.
They also look at defence lines during the war like the Seigfried Line, the Maginot Line, the Mareth Line, the Caesar Line etc. - and DON'T see the 3000+ pillboxes and bunkers built around London and in the Home Counties in the summer of 1940
Nor do they at first though remember that a British artillery-stiffened series of static defences was what defeated the DAK and the Italians in the first Seige of Tobruk
"Well, my days of not taking you seriously are certainly coming to a middle." - Malcolm Reynolds