An actual lawyer may want to jump in here, but the Redskins can still claim and defend the
Trademark "Redskins"
against infringement and dilution based on first use in there sphere of business (football, clothing, other merchandising). They don't need USPTO registration
to do that, although registration of a trademark makes it much simpler to defend in court.
The Redskins also have clear
Copyright ownership over their logo of the image of an Indian head in profile. In the US these days, all images are essentially automatically copyrighted by the author of the design. Registration of the copyright isn't necessary... and in my understanding is less needed to defend a original image from copyright violations in courts than trademark registration is useful for wordmarks.
A case may have additional strength if someone is using it with their Burgundy and Gold colors, particularly if used in combination with the term "Redskins." They can even probably go after "Washington" in those colors as dilution. The intent of the knockoff manufacturer would be pretty clear, IMO.
For reference, see the Steelers
suing and winning against Turtle Creek Sportswear's "generic" black & gold "Pittsburgh football" gear.
Even if the Redskins lose the appeals, I don't necessarily think the logo or nickname goes away. It seems that NFL Properties would still be able to defend against Redskin knockoffs fairly easily.