So, suppose the Brits exit the war in 1940...

I've been reading the May/June cabinet meetings etc in detail, and at that time Britain is sort of tied to its French ally, trying to keep a united front. There's at one point a lot of toing and froing over whether Mussolini will a) accept an appeasement deal to stay out of the war or even b) host a European peace conference. Reynaud is trying to keep the French in the fight, though Daladier and others are desperate to buy off Italy. Mussolini however wants his moment of glory and is intent on declaring war, so Ciano on his behalf is rebuffing any attempt to buy Italy off (IRRC he says that even if France offered Tunis, Algiers, Corsica and Nice it won't matter because Mussolini wants his war).

Given this, Halifax's hopes of getting something through his dealings with Bastianini (Italian ambassador to the UK) are probably forelorn, as are Hitler's hopes, if it was the case, to use the pause before Dunkirk as an opportunity for Britain to open negotiations.

Now, there are two possible ways for Dunkirk to go worse
1) No Pause - panzer units were already across the canal, and in range of Dunkirk when they were ordered back to an arbitrary line, and then sat there for 3 days whilst the BEF sorted itself out, and shored up its defences, especially along the canal. The decision had already been made in principal to withdraw to the coast, so no Pause wouldn't result in a renewed British push to link up with Weygand's largely theoretical offensive. The German fears of outrunning the infantry are irrelevant if the BEF is going backwards, not forwards. The result of no Pause could thus be that Gort does not get the chance to shore up his lines and make an orderly fighting retreat, but has to make the best of a disorganised one, at the same time facing panzers coming at him along the coast. It seems unlikely that the Belgians will fight on any longer than OTL and the marooned French army never looks capable of attacking even if the Germans are spread thinly. Its highly possible that the remaining Allied forces will be split into pockets, more so than OTL when half the French got cut off. If the panzers can break through Gort's rear he might not be able to even withdraw his main force on Dunkirk, let alone hold enough of the salient to get his men off the beaches. The RN was already evacuating as these things occurred, so some are going to get off, but with the panzers pressing the attack it might not be more than 20,000.

2) Operation Dynamo fails. Even the best estimates that Churchill's cabinet were believing was around 50,000 and that more were got off the beaches was due to the RAF driving off the Luftwaffe, and Gort's well-managed fighting retreat. You can absent the former if Dowding gets his way and the fighters are not released. You can absent the latter if Gort or his chief of staff are killed or injured in the fighting. Either way, you get far fewer people off the beaches because either the ships are sunk on the way in, or the front collapses earlier than OTL
 
Churchill was keeping morale high during the critical moments by issuing circulars instructing that all government minsters and departments give out positive vibes, and making speeches in the Commons that especially cheered the Labour and Liberal benches. He was carrying his junior ministers, and had most trouble with Halifax and with Chamberlain. He was able to carry the day initially because it became obvious that Mussolini was not seriously considering any deals, and because it was obvious that Hitler could use any formal requests aimed at the Italians on the greater issues to break down British morale. If it became known that Britain and Germany, however indirectly, were negotiating then the point of staying in longer would be lost.

A doomed BEF would result in over 100,000 British Prisoners of War being in German hands. The Fall of France is becoming inevitable - don't forget that the BEF hadn't done a great deal to avert that, its fighting being largely in a theatre of its own, albeit with the French Northern armies and the Belgians. But the Germans were continuing to advance on Paris, and Weygand was flubbering away trying to organise an offensive but not managing it.

What everyone was expecting after the Fall of France was an air war on Britain - that the aircraft factories would be wiped out and that civilians in the cities would be slaughtered. It was very much with this in mind that Halifax was talking in cabinet about a negotiated peace.

If only a few tens of thousands had been gotten off the beaches at Dunkirk, then the huge morale boost of the Evacuation would have been replaced by the worst defeat in British history, as Churchill was also calling the likelihood of it in cabinet meetings.

I think there would have been a huge upswell in calls for a JOINT negotiated peace to include France and Britain. This is something we forget looking back, but was something very much in their minds in May/June 1940, that the two allies had agreed not to make a separate peace and that what we know happened to France in OTL was not inevitable.

Regarding British concessions, Churchill said somewhere that if he thought he could get out of the war with only ceding Gibraltar, Malta and Suez then he would jump at the chance, but that he did not believe that was possible.

All of Churchill's bulldog spirit, his calls for a positive unified front, and speeches cheered in the Commons cannot make up for the loss of the BEF. When it becomes evident that France is about to fall, Halifax and Chamberlain (who, the minutes show, kept in the middle between Churchill and Halifax) would press Churchill to explore a joint peace offer with Reynaud, or his successor.
 
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