What I do know is that lathes operate at 100-1000 rpm.
Well I guess my idea just went in the trash cause I looked up my router and it is 8k-24k rpm,JR
I have no experience trying this with a router as that does indeed spin very fast. Which means really small items at best. Safety might the real thing to look out for though with possibly pressing a tool to do work well beyond what it was designed to do and withstand. (The workpieces might be heavier, longer, less straight, worse grip than compared to the normal router tools. A long and slightly out of center workpiece might loosen, bend or whip due to centrifugal forces and come looking for you in a hurry). You might instead consider using the router as a tool post grinder and have something else spin the workpiece against it. That way the router is actually used as a router with router tools, but you still get to turn with it. (I've done similar with a battery Dremel and a magnetic drill at a few occasions in off site desperation). As for speeds RPM is just part of the equation. Surface cutting speed is the key and that depends on diameter of the workpiece spinning as well as RPM. The optimal surface cutting speed varies for different types of materials, cutting tools used, depth of cut, machine power, how stiff and rigid your setup is and so on. (As for RPM reference I believe my little metal lathe goes to about 3000 RPM and takes workpieces up to about 8" diameter. The big lathe tops out at about 5500 RPM - it is much heavier and stiffer machine though). The best is likely just finding a nice used lathe. Some great used ones can sometimes be had for a song and a dance.
I've seen affordable adapters for drills...to make them small-purpose lathes...
Unlike a router a drill wont be made for taking much forces sideways, and most drills will likely be wobbling a fair bit around the center. After all their business is mostly about pressing straight down. One thing to consider is that with a machine you often end up with a fair collection of accessories and tools. They often cost more than the machine itself. If you make the machine yourself it might be an idea to make it accept accessories and tools from some existing machine or standard. Otherwise you might have to make all those too.
Either that or just find a broken drill press and mount it sideways?JR