As I recall, the elements he drew on from The End of Eternity took some liberties with the events of the book (like redefining what the end goal of the time travel was), so it was questionable whether it was really meant to be in continuity or if he was just borrowing some of its ideas and using them in a new context. But then, TEoE is a book about changing history, so maybe it's just an alternate version of events.
Asimov himself admitted that the reference to EoE in one of the later Foundation books wasn't consistent with that book. There is nothing in EoE about picking a timeline with no aliens. But EoE ends with a future that leads to the Galactic Empire and no more Eternity (so no more time changes). So it was already connected to the Empire/Foundation universe.