If an RTOS is able to run on one particular implementation of a CPU core, does it necessarily support other implementations of that core?

Typically, an RTOS, or, more specifically, a real-time kernel, is ported to a CPU architecture, not a particular chip.  In the case of uC/OS-III from Micrium, for example, there is a single kernel port for all Cortex-M3-based MCUs.  All that a developer needs, beyond this port, in order to run the kernel on a particular Cortex-M3 is start-up code--the same start-up code, essentially, that would be needed for any other type of C program.  Thus, although RTOS vendors like Micrium are not able to provide example projects for all of the numerous Cortex-M3-based platforms that exist, port code is available to enable developers to quickly get started on any such platform. 

Typically, an RTOS, or, more specifically, a real-time kernel, is ported to a CPU architecture, not a particular chip.  In the case of uC/OS-III from Micrium, for example, there is a single kernel port for all Cortex-M3-based MCUs.  All that a developer needs, beyond this port, in order to run the kernel on a particular Cortex-M3 is start-up code--the same start-up code, essentially, that would be needed for any other type of C program.  Thus, although RTOS vendors like Micrium are not able to provide example projects for all of the numerous Cortex-M3-based platforms that exist, port code is available to enable developers to quickly get started on any such platform.