This, I feel, has been a topic of long debate. Of course, there is no official point of separation as to when Earth-Two stories stopped being told and Earth-One stories started, because the writers weren't even aware of the concept until it was created in 1961. So some point in the 1950s seems to be the case. But when would be a good time to call it, based on the available evidence?
I've heard it said that the point where Batman's symbol changed to include the yellow oval is an accepted time, but I've also heard that refuted.
Batwoman and Bat-Girl became associated with Earth-One, but apparently they have Earth-Two versions as well. That doesn't mean any of the stories we saw them in back then were on Earth-Two. They may simply exist there as well, in untold stories.
It's funny because even though the Golden Age has not been considered canon for the Batman in currently-running comics since 1961, references to that Batman are still rife in comics, such as including the original Batmobile in the Batcave.
Was there ever anything shown of Earth-Two's post-Golden-Age adventures, such as him riding the later Batmobiles, that would shed some light on this? Certainly, I think that BY the time of the yellow oval, Earth-Two stories had stopped, if not at that point precisely.