It was possible. Invading the UK through gaining air superiority/naval superiority was possible, wasn't the pipe dream it was in WWI because of planes and RN's size being scaled down so much(and split in three). The issue was the Nazi's didn't care, it was not a priority for them. For example switching from trying to defeat the RAF to civilian bombing can be seen as stupid from our perspective if we're thinking of them getting air superiority to launch Operation Sealion but if the goal was to get the UK to quit that is a pretty rational switch to make(inspired by Berlin being bomed or not). The Nazi's goal was to acquire "living space" and the UK is an island whose power far outweighed her tiny size. The point of fighting the UK was to make the UK leave the war while expending as few resources as possible, before Hitler declared war on the US which paved the way for Torch, the Invasion of Italy and eventually D-Day it made much more tactical sense to just do what they did in 1941 and pretend the UK didn't exist while trying to destroy the USSR. The UK's main opponent at this point in the war were the Italians.What would have changed if Germany did win the Battle of Britain? It was possible. The British bombing raid of August 26th may have been stopped by weather and/or defenses. The Germans may not have over-reacted to the limited damage from that raid and continued their campaign against the RAF. Something else??
The Germans would have (short term) air superiority over Britain, but the factories turning out Spitfires, Hurricanes...... were out of bomber range and in a few months replacement aircraft and pilots would have become available. The Luftwaffe losses in men and aircraft were substantial and the German air forces lacked heavy bombers, moderating their ability to inflict sever damage to the island.
German nautical resources were hugely deficient in their ability to protect an invasion from the Royal navy. An invasion would likely have been mauled in the channel, at the loss of a number of Royal navy ships. Whatever losses the Royal navy suffers would have been counterbalanced by the fact that the Germans probably would have lost nearly every vessel capable of transporting troops across the channel with limited ability to replace these boats in short or medium term, by which time the RAF would probably be reconstituted.
Even if the Germans managed to successfully land some ground forces, armor and supply were huge issues to overcome. Living off the land was not an option in the Blitzkrieg tactical doctrine.
It appears to me that the outcome of the Battle of Britain is over-hyped and nothing would have changed if it historically went the other way.
And the UK not being a priority was reflected in Germany's pre WWII budgeting decisions where pretty much an entire navy was aborted during construction, the would be difficulty of Sealion in 1940/1941 is largely a consequence of that. The UK and all Europeans largely did the same but the UK had more ships in the first place.