[Jack]

There should be no accuracy difference-many of the world benchrest records are set with 20" benchrest barrels.
I would guess that a 7MM Rem Mag in a 20" barrel would give you a very loud, expensive, hard kicking 280 Rem, 7mm-08 or 7x57, in terms of performance.
The 338 Win has a less extreme expansion ratio (ratio of powder charge to bore diameter) than many cartridges labeled magnum. Consequently, the loss in velocity per inch of barrel cut off might not be quite as bad. Still, I suspect you would get performance similar to a 338-06- and again, louder, harder kicking, and more expensive to feed.
I agree that magnum cartridges need longer barrels to burn all that powder- 24" or 26" makes sense. To me, at least, a 20" barreled magnum doesn't.