I tend to think that the public-private question is a distraction, as is the differences of laws allowing unionization in one state from another. You can't have a system that works one way for private schools and another way for public, nor one that works in the north only or in the south only. In the end whatever the system evolves into will be the same for everyone.
I think that some of the possible outcomes have been discussed above. One is that the big BCS football powers spin off their football programs and form sort on an NFL farm league, using the same facilities, hiring the same coaches, etc, probably even carrying the same team names. That will solve a lot of problems, and leave the players to unionize, if they want, and take on the owners of the new teams. This is kind of the direction those AQ schools want to go, anyway - they want to be free at last from the NCAA requirements, so something of this nature wouldn't surprise me at all.
With or without the above happening, mf hits on another possible/probable outcome. It quite possibly could spell the end to athletic scholarships as we know them, not just for football, but for all sports. A university can't afford to have "employees" in any sport, not just football, because that opens up some major cans of worms. Thus, if this ruling holds up, I think that it is inevitable that all athletic scholarships will quickly cease to exist.
I can imagine a new system evolving from what we have. With all athletic scholarships eliminated, and only need-based scholarships available, the AQ teams would be in a situation of more, not less, power. Right now they can recruit 85 players, after which the non-AQ teams have an advantage as a player must choose between walking on at AQ-school, or taking a scholarship to non-AQ school. With no athletic scholarships at any school, there is no reason not to go to AQ-school.
One final comment - we have to realize that there is big money at stake, and there are creative people around. Some novel solutions could pop up. For example, suppose that somehow a new fund were created, say, the "Evil Empire Athletic Fund", and it awards scholarship to needy students who happen to play football, and just coincidentally play for OSU? Since the Evil Empire Fund was independent of the University, the players would not be being "paid" by the University. Since the Fund was charitable in nature, and since the players were doing no work for the fund, they wouldn't be employees of the fund, either....
The box of surprises has just barely been opened. There is much more to come, and we can just watch to see what happens.