Oh there's no doubt that the British Army was too small and had suffered tremendous losses before evacuation at Dunkirk. But the Germans had no transports that could have crossed the Channel, especially when challenged by the Royal Navy. And the Royal Navy got an infusion of 50 destroyers from the US (in exchange for overseas basing rights). These weren't the latest and greatest ships, but they were symbolic of the US commitment to Britain. This preceded Lend Lease.

There were several necessary prerequisites for invading Britain. First was destroying the RAF. Without absolute air supremacy, no invasion could be contemplated.

Second was destruction of the Royal Navy. Despite some success in attacking Scapa Flow with U-boats, the Kriegsmarine was simply incapable of challenging the Home Fleet, much less the combined might of the overseas fleets. The only hope would be from Luftwaffe bombers, primarily JU-87 Stukas. Unfortunately for the Germans, these suffered great losses in the Battle of France, as well as during the initial stages of the Battle of Britain. The Home Fleet was a considerable force in 1940: 4 battleships, 3 battlecruisers, 2 aircraft carriers, 20 cruisers, 25 destroyers, 23 submarines, and numerous smaller craft.

Finally, once ashore, the invasion force would have to overcome the British Army and the Home Guard. While these forces were seriously depleted, the Empire could have brought enormous numbers of troops from India, Egypt, and other regions. That this was never seriously considered might indicate that the British Army wasn't as weak as Churchill would have liked to portray.

I loved reading Churchill's diaries as a kid, but later on realized how much of them were self-aggrandizing and not as accurate as one would hope. Still a fascinating insight into one of the most important leaders of the 20th Century.