Opperation Sealion was a very real thing to the British, in 1940. Looking back on it, it is easy for us to see that a coordinated amphibious assult by the Germans on Britain was quite impracticle, but the Brits sure as Hell didn't think this way at the time. In fact, an invasion was expected.
With no army to defead against an invasion, Britain would have been helpless, or so many in Britain thought at the time.
There are some glaring problems with your logic.
1. The troops that were evacuated
left most of their equipment behind. You have men, minimally armed out there fighting exceptionally well armed Wehrmacht, with artillery and Panzer's. Are the these British Troops, recently evacuated, supposed to take out the tanks with their umbrellas?
Now, you can
eventually re-arm these troops, and re-organize them, but you are not going to be able to do it immediately. The evacuations began in late June. The major attacks on airfields began in late August. You are not to have the evacuated BEF re-armed by that point.
2. You have some units in the UK that were not in France, and some of the BEF in the
west of France that did evacuate with their equipment. You also had the "Home Guard," not great, but something.
3. Third, there is the time factor. The Nazis cannot contemplate an invasion until after the fighting in France is concluded. In our time line, that was in late June 1940. Eliminating the BEF would have probably pushed that back a few weeks; it would tie up German troops and supplies that were needed elsewhere. Whereas Hitler looked to Britain in early August 1940 in OTL this would be Late August or early September under your scenario.
In Britain sitting on it's hands while this is going on? No. It has resources, the British Empire. It can start getting in Commonwealth troops and equipment to defend Britain (I think that was done in OTL but later).
Germany has to a number of things prior to an invasion.
1. Get craft together to land troops.
2. Begin by eliminating the RAF.
3. Eliminate, via air power, the RN in the English Channel; either sink them or get them out.
Hitler might also delay for a bit to see if he can persuade Franco to let him take Gibraltar. That was his next goal.
How long does he have to do all this? Until late October, because Winter is coming. It will be impossible to cross the Channel in Winter. After that he has to wait until late Spring of 1941. Now, at that point, he
might be able to launch an invasion, but the Britain will be far from defenseless.
Hitler simply can't stage an invasion of England in 1940, so the presence of the BEF in England has very little bearing on the matter.
Now, will Britain negotiate? Possibly, but they will not accept any peace that leaves the invasion beaches in the hands of a hostile power. Why? Simple because Nazis
can launch invasion next Spring. So long as the Wehrmacht is sitting on those beaches, and so long as they are not advancing on Canterbury, Britain will
not make peace.