The rebooted universe was centered around 2010 Earth. Its history wasn't necessarily identical to the old universe's (remember, all those things that got erased from time are still gone*), but present-day England was identical to the old one—except, of course, as modified by Amy's memories.
You could speculate that, if Amy hadn't remembered the Doctor back, history would have ended up even more different than it did. (If you assume that there's some kind of "conservation principle" that means history changes as little as possible to make just enough sense, without the Doctor it would have to change a lot more than with him.) But that's just speculation. (Plus, we don't even know how radically history changed. The fact that the Doctor called up the Brigadier implies that the UNIT stories still mostly happened in the rebooted universe, at least, but we don't know much more than that.)
On the other hand, it's a pretty safe bet that it wouldn't be safe for too much longer beyond 2010 without the Doctor (See "Turn Left" for inspiration.)
See tardis:Howling:If the Doctor never existed... for more discussion on this exact topic including high amounts of speculation at levels that would be inappropriate to repeat here on this wiki namespace.
- "All those things that got erased from time are still gone" is an incorrect statement. There is absolutely nothing stated anywhere on screen in any episode (prove me wrong) to support this and in fact the statement is contradicted by the fact Amy's parents and Rory Williams, two things that were erased from time, were restored. Never mind the Doctor. And Torchwood and Sarah Jane, too.