I'd imagine repelling an invasion would only swell British pride and strengthen their resolve to continue the war since such a victory would, in the early years where Allied victories were a very rare thing, be the first indication that, no, the Nazi military is not unstoppable.
As for the implications on other theaters during the war, I'd say the most crucial element affected would be the German landing forces, which consist of:
- the sixth army
- the ninth army
- the sixteenth army
The Sixth army (the original one, not the reformed 1943-45 one) ended up destroyed at Stalingrad, the Ninth army basically fought during the entire Eastern front (including crucial roles in Operation Mars and Kursk), and the Sixteenth army sieged Leningrad before ultimately being destroyed in the Courland pocket
So assuming that each of these land forces were annihilated during an attempted Sea Lion, Operation Barbarossa would probably have to be delayed in order to scrounge up additional forces (if it even gets underway again, but given that Hitler explicitly stated his intentions of conquering the USSR, it's probable that it would've occurred one way or another), allowing the Soviets time to recover from Stalin's purges and rebuild their defenses, making Barbarossa that much harder when it does come, and probably facilitating a Soviet steamroll earlier than OTL.
Either way, a botched Seal Lion shortens the war by, IMO, a non-insignificant margin.
The Third Reich really didn't have much of a surface navy worth mentioning, so I'm not going to bother expanding on that, but I do think any losses they do take with regards to the Luftwaffe would come to bite them eventually when Allied strategic bomber raids begin in earnest.
After the war, however, I'm not too sure that a botched Sea Lion would change the reality that WW2 had left the British empire pretty weak and effectively broke. As much as Sea Lion would be a failure, I'm not too sure about the idea that it would cause absolutely zero damage. Any damage it does do only weakens Britain further (mind you, not as much as it would weaken Nazi Germany) and makes decolonization even more of a sure thing. No amount of patriotic sentiment is going to change this reality, and I do think this reality, especially if Eden still attempts his Suez Canal takeover, will still lead Britain's relative loss of power and activity on the world stage.