I'm guessing (since this is what my RTOS did) that interrupts were used to preempt the running task, and the OS would need to explicitly save and restore CPSR/S

I'm guessing (since this is what my RTOS did) that interrupts were used to preempt the running task, and the OS would need to explicitly save and restore CPSR/SPSR (the arm32 processor flags) for the interrupted process, since odds are another process would be scheduled to run next, with it's own execution state and CPSR.