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
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