Not much. Linux is a vast, massive, bloat bin. It'll never boot up as fast as RISC OS, no matter what you do setting up drivers and handling libraries will cause problems, it'll never fit in a 4mb rom, and chopping awful X-Windows means there'd be no software for it. Basically, no matter what is done to it, it'd still be a rickity advanced-user-only bloated Linux install with a clone of the RISC OS gui and no software. I just don't see any Acorn computing experience being reproduced with Linux.
The ideal would be to have a small, lightweight, well-designed OS clone of RISC OS for ARM and x86 processors, much like what AROS is to AmigaOS. Something that could be used with both Acorns and one of those ARM dev boards and would flawlessly run native RISC OS software, and would be transparently identical to RISC OS for the user in all except the cog/acorn logo. This would mean people could effectively continue using 'RISC OS' without any further hardware and OS development from the likes of Microdigital and RISCOS Ltd, should they exit, for there will continue to be a supply of ARM dev boards for a while yet.