There once was once something called an "Infinite USB Memory Drive" that acted as a thumb drive that read and wrote data to a networked share over wifi.
They aren't selling those anymore, but their approach might be what you need - network share to handle the multi- device connected to same drive issue.
In your shoes, I'd be looking at a Raspberry Pi Zero W (the one with wireless) instead of an arduino. More computing options means more ways to solve your problems with the same hardware.