The following is a reworked and updated version of this old blog post:
At the moment, the main reason why English students are not all going to university in Scotland (where university tuition is free, compared to English universities that will typically charge £27,000 for a 3-year degree) is that Scottish universities charge them up to £27,000 for their degree. This is only possible because the EU rule about not discriminating against EU students only applies to students from other EU countries (such as Ireland, Denmark or Bulgaria) and not to students from other parts of the UK (England, Wales and Northern Ireland).
As soon as Scotland regains her independence, rUK students become EU students and will have to be treated in the same way as students from Scotland.
This is an area where the Scottish Government's White Paper is a bit vague, and many unionists have now started claiming that Scotland will have no choice but to introduce tuition fees after independence (see this article by Severin Carrell for details).
However, some lessons can be learnt from Scandinavia, where the closely related languages in theory make it easy for students to study in the other Nordic countries, and EU rules mean these foreign students can't be discriminated against based on citizenship.
Denmark used to have great problems in this area. For instance, large numbers of Swedes used to study medicine in Copenhagen and then go home straight after graduation. In 2007, Denmark therefore did two things (link in Danish): (1) They changed the number of advanced highers ("højniveaufag") a student needs to pass to get a grade top-up, which benefited Danes in comparison with Swedes. (2) They changed the way they translated Swedish grades into Danes ones (that is, they made it harder for them to get in).
Apart from this, Denmark pays generous grants (typically £7616 per year) to university students who were living in Denmark prior to starting university. (Denmark used to require students to have lived there for at least five years in order to qualify, but this is an area that the EU is currently clamping down on.)
Scotland could copy some of these policies after independence.
There are already plenty of differences between A Levels and Scottish Highers and Advanced Highers, so it would be easy to tweak the entry requirements to make it harder for rUK students to get into Scottish universities. Scotland could also introduce a new grading system different from the one used in the rUK, which would then need to be converted. The very best rUK students would of course still get in, but that would be to Scotland's advantage anyway. (The rUK might retaliate and make it harder for Scottish students to get into their universities, but you can't have your cake and eat it, too.)
Scotland could also introduce tuition fees for everybody, but cancel out the effect by creating grants for Scottish citizens and residents. However, as I wrote above, the EU is not too happy about creating too many restrictions in this area.
In an ideal world such measures shouldn't be necessary, but until it dawns on the English that they're shooting themselves in the foot by pricing bright young people out of universities, I fear that Scotland will have to take a leaf out of Denmark's book.
Finally, England is the odd man out in the EU when it comes to tuition fees. Most EU countries have either no fees or very low ones. Scotland might be able to convince the other countries that England's sky-high fees are distorting the free movement of students and that restrictions have to be placed on English students until England lowers its fees. This would be an ideal solution.