When a second processor (or "co-processor") was used through the BBC Micro's Tube, the Beeb's internal CPU and RAM was used for display and I/O purposes, whereas the main program code would be run on the CPU and RAM in the second processor's case. This code would include an operating system to suit the processor in question, e.g. CP/M on the Z80 second processor, DOS-plus on the 80186 coprocessor, etc.
The ARM Evaluation System was shipped in 1986, and as well as a ROM-based OS (like the Beeb) it was provided with software on six floppies. These included an assembler and debugger, some example utility programs, a port of BBC Basic, and LISP, PROLOG and FORTRAN.
As the ARM was a brand new processor, this was the first OS and software released for it. But rather than being related to RISC OS, it was running an early version of Acorn's ARX operating system:
By 1986, the full four-chip set was working, and a second processor version of the ARM was built for software development. 'We did a lot of software development both in the UK and the States, and a Xerox-like operating system was developed,' explains Wilson, 'This was called ARX. We had it working well on just a single chip - a windows system with VDU calls across the Tube.'
Quotation source.
The Archimedes line, with RISC OS's antecendent OS "Arthur", was released one year later in June 1987. Arthur was a rushed stop-gap OS for the A305 and A310 while ARX was being finished. When ARX was further delayed, a revised edition of Arthur (with a graphical interface) was released as RISC OS 2.0.
On that basis, it seems that no early version of RISC OS ran on the BBC Micro, but an early version of ARX did.
Do bear in mind that the software on the ARM Evaluation System, like the other second processors, ran on the external processor, and could only communicate with the Beeb through the window of the Tube. Some, such as the ARM E.S., had a ROM of software that was inserted into the Beeb, but these were effectively library functions that could be called by the ARM E.S. through the Tube. They were actually 8-bit code running natively on the Beeb's 6502 and MOS operating system.
There are more resources relating to the Arm Evaluation System, including documentation and copies of the on-board ROM, here.