Action at Poperinge
The route back from Brooke's position to Dunkirk passed through the town of Poperinge (known to most British sources as "Poperinghe"), where there was a bottleneck at a bridge over the Yser canal. Most of the main roads in the area converged on that bridge. On 27 May, the Luftwaffe bombed the resulting traffic jam thoroughly for two hours, destroying or immobilising about 80% of the vehicles. Another Luftwaffe raid—on the night of 28/29 May—was illuminated by flares as well as the light from burning vehicles. The 44th Division in particular had to abandon many guns and lorries, losing almost all of them between Poperinge and the Mont.
The German 6. Panzerdivision could probably have destroyed the 44th Division at Poperinge on 29 May, thereby cutting off 3rd Division and 50th Division as well. Thompson calls it "astonishing" that they did not, but they were distracted by investing the nearby town of Cassel.
What if the 6th Panzer Division did just this? What would it mean for the evacuation and defensive efforts to hold back the German advance?
I personally don't know enough about the particulars of the battle to make an informed comment, but it seems that in this case then the perimeter would have a major gap that would be exploited, stopping the evacuation early and potentially preventing most of the historical evacuees from escaping. If so this would potentially have serious political consequences, not to mention military ones in the near future. Despite the British war cabinet being aware of the disaster in France that was brewing and being prepared for it, the public was not and would not have been prepared emotionally for losing most of the BEF at Dunkirk. Historically the defeat of France was mitigated by the successful evacuation of most of the BEF sans equipment, but it was still a propaganda miracle that it was pulled off. But if it had failed or at least only resulted in some 50-75,000 men evacuated I think it would have been a much harsher blow to British national morale. I doubt it would make a long run difference in terms of the British surrendering, but in the short term if coupled with major defeats elsewhere, it could potentially hurt Churchill politically.
Any idea though what this would mean for the African Campaign if the BEF was not successfully evacuated?