Some historians would argue that Hitler had no interest in taking Britain. He long admired the British, at one point he even offered them control of the seas while he expanded into Eastern Europe and Russia.

On top of this, the Germans did not have the capability to invade Britain even if they wanted to. One of the major preconditions to them invading the mainland would be both naval and air-force superiority, which simply was not possible. Couple this with the logistical issues involved with this sort of invasion, and the previous points, it was never going to happen. Hitler expected the British to "sue for peace", which they didn't.

So, I do not believe the British ever truly were in a position where they would have needed to surrender. I do not think it is as close as it has been dramatized for decades after.