So the part that interests me is when the initial wave of landings start going badly, does the German high command fish or cut bait. I'm leaning towards the latter, as their OTL reactions towards Sea Lion were basically "NopeNopeNope", so it depends on how they get overruled in ATL. If they pull back quickly after say the first wave fails, they can get away with a pretty light loss of troops, although with severe beatings to the KM and Luftwaffe. But if they double down, it could really turn into a meat grinder.
I'm also of the opinion that any kind of setback from Sea Lion, mild or severe, would push back Barbarossa a year. If you want to drive to Moscow and St. Petersburg before the fall rains and winter freeze sets in, June is about the latest you can step off. It's probably too late as it was. So say we get a two month delay and you're looking at an August '41 launch date for Barbarossa; you either bank on your panzer armies being able to drive 800 miles in about 8 weeks before their roads crap out, or you push it back till spring/summer '42. Maybe Hitler is aggressive and paranoid enough to demand a late-summer offensive anyway, but my guess is the guys in OKH would prefer to wait. If they did step off later in '41, with a few less Panzers and Messerschmitts along with them, the invasion wouldn't go any better than OTL.
So let's say Barbarossa is pushed back till mid '42 - what does everyone do in 1941?
Germany has almost all of Europe under its thumb and Britain has no way to contest them there. Sea Lion has proven the Nazis can't reach across the Channel and touch them, but they're in no position to touch the Nazis. So you're looking at a stalemate as far as Europe is concerned; this may turn into a Phony War round 2.
One of the biggest butterflies from this is that Germany will have time to stockpile oil. With the Soviet Union still glad to sell that oil to them and no blitzkrieg armies chewing through barrels of it, Germany may be able to avoid the shortages that plagued them in OTL.
The US is probably going to get more aggressive with Germany after A. they tried to invade our ally, and B. they look beatable, after a huge blowout in France is followed a couple months later with a wipeout in the Channel. You might even see a 1940 declaration of war, although that's not going to do anything in the short term b/c see above.
My guess is the US approaches war in 1940 a little more casually than they did in OTL 1941. In OTL, Japan had stabbed them in the back, the Soviets were on the brink, and Britain was being bombed and blockaded, so there was definitely some sense of urgency as the Allied cause looked to be at a low ebb. In ATL, there's no Pacific war (although anyone can see that things are likely heading that way), the USSR isn't the "enemy of my enemy" yet, and Britain doesn't look to be in any imminent danger. So the Americans are probably going to slow-play any type of Lend Lease or proto-D Day build up.
The USSR is only going to keep building up; I'm not sure if the Winter War was enough of a kick in the ass to get their army in shape. My guess is the green post-Purge officers of OTL 1941 are going to just be green post-Purge officers in ATL 1942. But materially, they're going to be on stronger footing. I don't know if Stalin had any other adventures planned like Finland, but throughout his career, he was usually less focused on expansion than on holding what he had. So my guess is he stays back on the defensive, tightening his grip on Poland and amassing his army.
Italy - could a failed Sea Lion be enough of a butterfly to get Mussolini to sit the war out? That would only be a net positive for Germany, not to mention Italy itself. You'd have hundreds of thousands of young men still alive, rather than dying in faraway places like Stalingrad for some very questionable benefit to the nation.
Southeast Asia - the Royal Navy is going to be a lot more free to redeploy after I'm assuming the KM surface force is gutted in Sea Lion. Force Z might arrive with some friends, for example.