The LDV / Home Guard might have had numbers in mid to late 1940 but certainly lacked training, orgainzation, and equipment. In the early days most volunteers got just an arm band (uniforms came later). This made them legal combatants in the same way the Germans did this with the Volksturm towards the end of the war.
One also has to note the guns the US sent were the SMLE No 3 Mk I and I* (or P 14) both of which used .30 06 (7.62mm) ammunition not British .303 (7.92mm). To note this many had a white band painted around the stock about half way to the muzzle. You sometimes see this in Home Guard photos.
Certainly, a unit of even fairly well equipped Home Guard with say some Lewis guns, a BAR or three (yes they received some of these too), and a couple of Smith guns was hardly an awesome fighting force.

Now, a more interesting proposition is that the Germans do something totally different.

What if instead of launching an invasion, the Germans use their preparations to assist the U-boat campaign? The British held at a minimum 36 destroyers in the South of England for counter invasion duties. This in turn put a real strain on the Royal Navy's ability to escort convoys properly.
So, now the Luftwaffe turns its attentions to RN destroyers and other naval vessels in port and tries to attack them at sea as well. With the continued build up of shipping and other preparations for a landing the attacks would not look out of place.
At the same time priority is given to U-boat constuction and operations. If the British had a few more thrid to half million ton losses per month as they did in this period historically, it might have really put some economic hurt on England.

__________________
Truth is stranger than bullshit!