The standard Nano you have is using the Duemilanove bootloader, which works at 57600. Even though it's a different form factor, you program it by picking "Duemilanove" from the board type menu.
The Uno uses OptiBoot for its bootloader, which works at 115200 baud. It's the new standard because it's smaller (512 bytes vs about 2K). When you burned your new standalone chip you got OptiBoot.
So, if you treat your new Nano clone as a Uno, everything should work fine.
Good luck!