I have heard it said that inside the event horizon of a black hole, space is stretched to infinity (so you can never reach the singularity, but only fall towards it forever) and time is condensed, so if you fall into a black hole (and don't get vaporized by the accretion disk or ripped apart by tidal forces), you will see the universe outside the black hole rapidly age. There are hypotheses that white holes are born inside of black holes, in the infinite space therein, effectively new Big Bangs which give birth to new universes.So our cosmology is an unending tree of universes within universes, where black holes are the cradles.

So since black holes do have finite lifespans (though they are insanely long), if we live inside a chain of singularities-within-singularities, and time is moving extremely fast in the outer layers relative to us, what would happen to our universe when hawking radiation completely dissipates the outer layer black holes?

I can't help but feel like I have a major misunderstanding of how this all works.