Well, if the BEF is annihilated and the RAF takes a fair bit of damage fighting the Luftwaffe in France, I think Britain might well end up making peace, though that will probably require Churchill getting tossed out first.
If Britain does decide to keep fighting, they'll have a much harder time of it. The BEF being destroyed mean that they'll have a serious shortage of trained soldiers, which means that the North African campaign probably won't be as bad an embarrassment for Italy as it was in OTL, and probably results in Britain having to leave their Pacific possessions with even less than they had in OTL to make up for the lost units of the BEF.
At the least, the Battle of Britain is going to be a lot longer and much more painful for the UK, with Britain maybe not regaining control of their home skies until Hitler shifts the Luftwaffe out for Barbarossa. If control of the skies makes Hitler too bold he might even try an invasion or an attack with airborne troops, though I doubt he could manage a successful full-scale invasion and occupation.
Interestingly, does anyone think Britain being in a much more desperate situation might prompt an earlier US entry into the war? There was clearly a fair bit of sympathy for Britain in the US, as Lend-Lease and such indicates, might Britain being invaded, even if unsuccessfully, be enough to overcome the isolationist sentiment?
Also, from what I recall Churchill threatened to use poison gas on the Germans if they invaded Britain. Certainly I can't see Britain just being invaded and occupied while they still have fairly large stockpiles of mustard gas that haven't been used. Presumably the Germans will respond in kind if Britain gasses their troops, and given how such things work I'd imagine before long everyone will be using poison gas just as happened in WWI. Anyone care to speculate on how that will effect the course of the war?