It would be happening at the same time, sure. If the universe is infinite, then somewhere out there, there must, by the laws of infinity, exist worlds that are EXACTLY the same as Earth in every way, just 5 minutes in the past.
You have to understand that compared to how many other arrangements there will be, these arrangements will be so rare they'll be near impossible to find... say you are in a spaceship that can jump to random places in our infinite universe and just take a snapshot of whats up, 99.99999999999999999% of those will just be boring space.... but a small, incredibly small, proportion of those snapshots, will show, given enough, humans.
Theres a difference between a infinite universe and a "field" of universes, or Multiverses. You could theoretically travel to these other places in our infinite universe (if you had some sort of ship that could go faster than light, but thats fucked for a number of reasons, so just hypothetically). For multiverses, those are other universes that are unreachable as far as we know - you can think of them as separate spheres, some with different laws of physics, etc. No matter how far you travel in one of these, you will never reach another "multiverse", as they're all unconnected.