Hm, this is one of those topics that gets some folk say "impossibru!!!" no matter what. Of course, real history happened the way we know (so should be grateful for that). However, it doesn't take much for things to have been different, and no, it doesn't need any volcanoes. Have the german leadership be a little more farsighted in 1939-early 1940 and actually seriously plan for the possiblity that the UK won't accept an armistice. Have them fit drop tanks to the Bf-109E and 110 earlier, and give a bit more attention to actually combating Royal Navy in such a case (an earlier Fliegerkorps X if you like.)
To have them proceed with the invasion means they have achieved air superiority over the landing zones, no doubt the losses during the landing would have been heavy, but once on shore the german tanks and infantry would be unstopable imo, i don't think anyone could stop them in 1940 without a large numerical superiority (think USSR). The best outcome would be a sort of France situation for UK, i.e. one part under german control and the rest "independent".
A POD closer to the invasion date would be if Hitler doesn't order the switch to bombing London and LW keeps up the pressume on Fighter Command. As i understand it, they were close to breaking point. The switch to just bombing London was a godsend. The above apply in this case as well, although the chances of success are reduced. But there is still a far from insignificant chance the invasion might still succeed.
As i'm very curious about the real situation in September 1940 in the UK, are there any details about where the RN ship were deployed, how many and how far from the invasion area, how many tanks they had, guns, troops, that kind of details?
Apart from Fighter Command, afaik Bomber Command didn't had more than 500 aircraft operational at that time, no doubt they will all be bombing the beaches, but have a look at the gigantic number of aircraft and sorties needed to reduce the combat ability of the german troops in places like France 1944. The situation is different in 1940, german troops would be exposed, but compared to 1942 let alone 1944, i think Bomber Command's effect would me much less than it is assumed. Of course they would also throw at the beaches every other available older or newer aircraft that can be armed and machine-gun the germans, but they will likely be slaughtered by flak.
Speaking of gas attacks, say the british actually do that during a hypothetical invasion, do the germans have stockpiles of gas of their own to retaliate?
Finally yeah, regarding losses, in a succesful invasion 3 times the losses suffered during the french campaign sounds about right imo, which actually would be just a pinprick compared to the losses suffered in the OTL Barbarossa campaign.