I can't find any history on what Acorn used as the development environment for early versions of RISC OS, but a BBC Micro with an ARM second processor seems possible. Was any early form of an ARM-specific OS ever made public for this hardware setup?
This page describes one of the prototype A500s used by Paul Fellows (who led the team in charge of developing Arthur, the operating system which eventually became RISC OS). Paul Fellows himself said
This machine is the one I used for development of the Operating System at Acorn. Originally these machines were hooked up to BBC micros via an umbilical into the tube port and all the I/O was done by the Beeb. Over time we gradually got the various sub-systems alive, and moved them over to the A500 native. First the video and graphics, then the keyboard, then the file systems and so on. This machine was on my desk when for the very first time ever we removed the umbilical, and rebooted it - and watched it come up to the basic prompt on its own. Arthur was born!
So Arthur never ran on a BBC Micro, at least, not using the 6502 as the main CPU, but a BBC Micro was necessary at first to provide the I/O (as you’d expect from a system using the tube).
Arthur was released to the public with the A305 and A310. I don’t think a BBC Micro-using version was ever released. It was supposed to be a stop-gap operating system until ARX was completed, but when the latter failed to materialise, development of Arthur was extended and it became RISC OS with the release of version 2.
All this is described in more detail on the dedicated Wikipedia page.
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.'
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.