You need the MMU version of the Nios2 CPU to run Linux, and even then IIRC it's a heavily patched version of Linux, not an official one. And it runs rather slowly. A smaller real time OS optimized for embedded platforms, such as FreeRTOS, could be a better choice. Those usually don't even require an MMU.
As you said the free version of Nios doesn't have an MMU. You can still make your own project with the full Nios CPU with MMU, but then it will run in "opencore evaluation mode", meaning that the JTAG needs to be tethered to your PC while you run your application. As soon as you disconnect the USB port doing the JTAG communication (of after a fixed time delay after that) the Nios CPU stops working.